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Full text of "Classical Electrodynamics"

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Classical Electrodynamics 




by JOHN DAVID JACKSON 

Professor of Physics, University of Illinois 



CLASSICAL 
ELECTRODYNAMICS 



John Wiley & Sons, Inc., New York • London • Sydney 




ii febi-axov- 







20 19 18 17 16 15 14 

Copyright © 1962 by John Wiley & Sons, Inc. Ail rights reserved. 
This book or any part thereof must not 
be reproduced in any form without the 
written permission of the publisher. 

Printed in the United States of America 
Library of Congress Catalog Card Number: 62-8774 

ISBN 471 43131 1 



To the memory of my father, 
Walter David Jackson 



Preface 



Classical electromagnetic theory, together with classical and quan- 
tum mechanics, forms the core of present-day theoretical training for 
undergraduate and graduate physicists. A thorough grounding in these 
subjects is a requirement for more advanced or specialized training. 

Typically the undergraduate program in electricity and magnetism 
involves two or perhaps three semesters beyond elementary physics, with 
the emphasis on the fundamental laws, laboratory verification and elabora- 
tion of their consequences, circuit analysis, simple wave phenomena, and 
radiation. The mathematical tools utilized include vector calculus, 
ordinary differential equations with constant coefficients, Fourier series, 
and perhaps Fourier or Laplace transforms, partial differential equations, 
Legendre polynomials, and Bessel functions. 

As a general rule a two-semester course in electromagnetic theory is 
given to beginning graduate students. It is for such a course that my book 
is designed. My aim in teaching a graduate course in electromagnetism is 
at least threefold. The first aim is to present the basic subject matter as a 
coherent whole, with emphasis on the unity of electric and magnetic 
phenomena, both in their physical basis and in the mode of mathematical 
description. The second, concurrent aim is to develop and utilize a number 
of topics in mathematical physics which are useful in both electromagnetic 
theory and wave mechanics. These include Green's theorems and Green's 
functions, orthonormal expansions, spherical harmonics, cylindrical and 
spherical Bessel functions. A third and perhaps most important pur- 
pose is the presentation of new material, especially on the interaction of 



VIII 



Preface 



relativistic charged particles with electromagnetic fields. In this last area 
personal preferences and prejudices enter strongly. My choice of topics is 
governed by what I feel is important and useful for students interested in 
theoretical physics, experimental nuclear and high-energy physics, and that 
as yet ill-defined field of plasma physics. 

The book begins in the traditional manner with electrostatics. The first 
six chapters are devoted to the development of Maxwell's theory of 
electromagnetism. Much of the necessary mathematical apparatus is con- 
structed along the way, especially in Chapters 2 and 3, where boundary- 
value problems are discussed thoroughly. The treatment is initially in 
terms of the electric field E and the magnetic induction B, with the derived 
macroscopic quantities, D and H, introduced by suitable averaging over 
ensembles of atoms or molecules. In the discussion of dielectrics, simple 
classical models for atomic polarizability are described, but for magnetic 
materials no such attempt is made. Partly this omission was a question of 
space, but truly classical models of magnetic susceptibility are not possible. 
Furthermore, elucidation of the interesting phenomenon of ferromagnetism 
needs almost a book in itself. 

The next three chapters (7-9) illustrate various electromagnetic pheno- 
mena, mostly of a macroscopic sort. Plane waves in different media, 
including plasmas, as well as dispersion and the propagation of pulses, are 
treated in Chapter 7. The discussion of wave guides and cavities in Chapter 
8 is developed for systems of arbitrary cross section, and the problems of 
attenuation in guides and the Q of a cavity are handled in a very general 
way which emphasizes the physical processes involved. The elementary 
theory of multipole radiation from a localized source and diffraction 
occupy Chapter 9. Since the simple scalar theory of diffraction is covered 
in many optics textbooks, as well as undergraduate books on electricity and 
magnetism, I have presented an improved, although still approximate, 
theory of diffraction based on vector rather than scalar Green's theorems' 
The subject of magnetohydrodynamics and plasmas receives increasingly 
more attention from physicists and astrophysicists. Chapter 10 represents 
a survey of this complex field with an introduction to the main physical 
ideas involved. 

The first nine or ten chapters constitute the basic material of classical 
electricity and magnetism. A graduate student in physics may be expected 
to have been exposed to much of this material, perhaps at a somewhat 
lower level, as an undergraduate. But he obtains a more mature view of it, 
understands it more deeply, and gains a considerable technical ability in 
analytic methods of solution when he studies the subject at the level of this 
book. He is then prepared to go on to more advanced topics. The 
advanced topics presented here are predominantly those involving the 



Preface ix 

interaction of charged particles with each other and with electromagnetic 
fields, especially when moving relativistically. 

The special theory of relativity had its origins in classical electrodynamics. 
And even after almost 60 years, classical electrodynamics still impresses 
and delights as a beautiful example of the co variance of physical laws under 
Lorentz transformations. The special theory of relativity is discussed in 
Chapter 11, where all the necessary formal apparatus is developed, various 
kinematic consequences are explored, and the covariance of electrodynamics 
is established. The next chapter is devoted to relativistic particle kine- 
matics and dynamics. Although the dynamics of charged particles in 
electromagnetic fields can properly be considered electrodynamics, the 
reader may wonder whether such things as kinematic transformations of 
collision problems can. My reply is that these examples occur naturally 
once one has established the four- vector character of a particle's momentum 
and energy, that they serve as useful practice in manipulating Lorentz 
transformations, and that the end results are valuable and often hard to 
find elsewhere. 

Chapter 13 on collisions between charged particles emphasizes energy 
loss and scattering and develops concepts of use in later chapters. Here 
for the first time in the book I use semiclassical arguments based on the 
uncertainty principle to obtain approximate quantum-mechanical ex- 
pressions for energy loss, etc., from the classical results. This approach, so 
fruitful in the hands of Niels Bohr and E. J. Williams, allows one to see 
clearly how and when quantum- mechanical effects enter to modify classical 
considerations. 

The important subject of emission of radiation by accelerated point 
charges is discussed in detail in Chapters 14 and 15. Relativistic effects 
are stressed, and expressions for the frequency and angular dependence of 
the emitted radiation are developed in sufficient generality for all appli- 
cations. The examples treated range from synchrotron radiation to 
bremsstrahlung and radiative beta processes. Cherenkov radiation and the 
Weizsacker- Williams method of virtual quanta are also discussed. In the 
atomic and nuclear collision processes semiclassical arguments are again 
employed to obtain approximate quantum-mechanical results. I lay con- 
siderable stress on this point because I feel that it is important for the 
student to see that radiative effects such as bremsstrahlung are almost 
entirely classical in nature, even though involving small-scale collisions. 
A student who meets bremsstrahlung for the first time as an example of a 
calculation in quantum field theory will not understand its physical basis. 

Multipole fields form the subject matter of Chapter 16. The expansion 
of scalar and vector fields in spherical waves is developed from first 
principles with no restrictions as to the relative dimensions of source and 



x Preface 

wavelength. Then the properties of electric and magnetic multipole radia- 
tion fields are considered. Once the connection to the multipole moments 
of the source has been made, examples of atomic and nuclear multipole 
radiation are discussed, as well as a macroscopic source whose dimensions 
are comparable to a wavelength. The scattering of a plane electromagnetic 
wave by a spherical object is treated in some detail in order to illustrate a 
boundary- value problem with vector spherical .waves. 

In the last chapter the difficult problem of radiative reaction is discussed. 
The treatment is physical, rather than mathematical, with the emphasis on 
delimiting the areas where approximate radiative corrections are adequate 
and on finding where and why existing theories fail. The original Abraham- 
Lorentz theory of the self-force is presented, as well as more recent classical 
considerations. 

The book ends with an appendix on units and dimensions and a biblio- 
graphy. In the appendix I have attempted to show the logical steps 
involved in setting up a system of units, without haranguing the reader as 
to the obvious virtues of my choice of units. I have provided two tables 
which I hope will be useful, one for converting equations and symbols and 
the other for converting a given quantity of something from so many 
Gaussian units to so many mks units, and vice versa. The bibliography 
lists books which I think the reader may find pertinent and useful for 
reference or additional study. These books are referred to by author's 
name in the reading lists at the end of each chapter. 

This book is the outgrowth of a graduate course in classical electro- 
dynamics which I have taught off and on over the past eleven years, at both 
the University of Illinois and McGill University. I wish to thank my 
colleagues and students at both institutions for countless helpful remarks 
and discussions. Special mention must be made of Professor P. R. Wallace 
of McGill, who gave me the opportunity and encouragement to teach what 
was then a rather unorthodox course in electromagnetism, and Professors 
H. W. Wyld and G. Ascoli of Illinois, who have been particularly free with 
many helpful suggestions on the treatment of various topics. My thanks 
are also extended to Dr. A. N. Kaufman for reading and commenting on a 
preliminary version of the manuscript, and to Mr. G. L. Kane for his 
zealous help in preparing the index. 

J. D. Jackson 

Urbana, Illinois 
January, 1962 



Contents 



chapter 1. Introduction to Electrostatics 

1.1 Coulomb's law, 1 . 

1.2 Electric field, 2. 

1.3 Gauss's law, 4. 

1 .4 Differential form of Gauss' s law, 6. 

1.5 Scalar potential, 7. 

1.6 Surface distributions of charges and dipoles, 9. 

1.7 Poisson's and Laplace's equations, 12. 

1.8 Green's theorem, 14. 

1.9 Uniqueness theorem, 15. 

1.10 Formal solution of boundary-value problem, Green's functions, 18. 

1.11 Electrostatic potential energy, 20. 

References and suggested reading, 23. 
Problems, 23. 



chapter 2. Boundary-Value Problems in Electrostatics, I 26 

2.1 Method of images, 26. 

2.2 Point charge and a grounded conducting sphere, 27. 

2.3 Point charge and a charged, insulated, conducting sphere, 31. 

2.4 Point charge and a conducting sphere at fixed potential, 33. 

2.5 Conducting sphere in a uniform field, 33. 

2.6 Method of inversion, 35. 

2.7 Green's function for a sphere, 40. 



xii Contents 

2.8 Conducting sphere with hemispheres at different potentials, 42. 

2.9 Orthogonal functions and expansions, 44. 

2.10 Separation of variables in rectangular coordinates, 47. 

References and suggested reading, 50. 
Problems, 51. 

chapter 3. Boundary-Value Problems in Electrostatics, II 54 

3.1 Laplace's equation in spherical coordinates, 54. 

3.2 Legendre polynomials, 56. 

3.3 Boundary-value problems with azimuthal symmetry, 60. 

3.4 Spherical harmonics, 64. 

3.5 Addition theorem for spherical harmonics, 67. 

3.6 Cylindrical coordinates, Bessel functions, 69. 

3.7 Boundary-value problems in cylindrical coordinates, 75. 

3.8 Expansion of Green's functions in spherical coordinates, 77. 

3.9 Use of spherical Green's function expansion, 81. 

3.10 Expansion of Green's functions in cylindrical coordinates, 84. 

3.11 Eigenfunction expansions for Green's functions, 87. 

3.12 Mixed boundary conditions, charged conducting disc, 89. 
References and suggested reading, 93. 

Problems, 94. 

chapter 4. Multipoles, Electrostatics of Macroscopic Media, 

Dielectrics 98 

4.1 Multipole expansion, 98. 

4.2 Multipole expansion of the energy of a charge distribution in an 
external field, 101. 

4.3 Macroscopic electrostatics, 103. 

4.4 Simple dielectrics and boundary conditions, 108. 

4.5 Boundary-value problems with dielectrics, 1 10. 

4.6 Molecular polarizability and electric susceptibility, 116. 

4.7 Models for molecular polarizability, 119. 

4.8 Electrostatic energy in dielectric media, 123. 
References and suggested reading, 127. 
Problems, 128. 

chapter 5. Magnetostatics 132 

5.1 Introduction and definitions, 132. 

5.2 Biot and Savart law, 133. 

5.3 Differential equations of magnetostatics, Ampere's law, 137. 

5.4 Vector potential, 139. 

5.5 Magnetic induction of a circular loop of current, 141. 

5.6 Localized current distribution, magnetic moment, 145. 



Contents xiii 

5.7 Force and torque on localized currents in an external field, 148. 

5.8 Macroscopic equations, 1 50. 

5.9 Boundary conditions, 154. 

5.10 Uniformly magnetized sphere, 156. 

5.11 Magnetized sphere in an external field, permanent magnets, 160. 

5.12 Magnetic shielding, 162. 
References and suggested reading, 164. 
Problems, 165. 

chapter 6. Time- Varying Fields, Maxwell's Equations, Con- 
servation Laws 169 

6.1 Faraday's law of induction, 170. 

6.2 Energy in the magnetic field, 173. 

6.3 Maxwell's displacement current, Maxwell's equations, 177. 

6.4 Vector and scalar potentials, wave equations, 179. 

6.5 Gauge transformations, 181. 

6.6 Green's function for the time-dependent wave equation, 183. 

6.7 Initial- value problem, Kirchhoff's integral representation, 186. 

6.8 Poynting's theorem, 189. 

6.9 Conservation laws, 190. 

6.10 Macroscopic equations, 194. 
References and suggested reading, 198. 
Problems, 198. 

chapter 7. Plane Electromagnetic Waves 202 

7.1 Plane waves in a nonconducting medium, 202. 

7.2 Linear and circular polarization, 205. 

7.3 Superposition of waves, group velocity, 208. 

7.4 Propagation of a pulse in a dispersive medium, 212. 

7.5 Reflection and refraction, 216. 

7.6 Polarization by reflection, total internal reflection, 220. 

7.7 Waves in a conducting medium, 222. 

7.8 Simple model for conductivity, 225. 

7.9 Transverse waves in a tenuous plasma, 226. 
References and suggested reading, 231. 
Problems, 231. 

chapter 8. Wave Guides and Resonant Cavities 235 

8.1 Fields at the surface of and within a conductor, 236. 

8.2 Cylindrical cavities and wave guides, 240. 

8.3 Wave guides, 244. 

8.4 Modes in a rectangular wave guide, 246. 

8.5 Energy flow and attenuation in wave guides, 248. 



civ Contents 

8.6 Resonant cavities, 252. 

8.7 Power losses in a cavity, 255. 

8.8 Dielectric wave guides, 259. 
References and suggested reading, 264. 
Problems, 264. 

chapter 9. Simple Radiating Systems and Diffraction 268 

9.1 Fields and radiation of a localized source, 268. 

9.2 Oscillating electric dipole, 271. 

9.3 Magnetic dipole and quadrupole fields, 273. 

9.4 Center-fed linear antenna, 277. 

9.5 Kirchhoff's integral for diffraction, 280. 

9.6 Vector equivalents of Kirchhoff's integral, 283. 

9.7 Babinet's principle, 288. 

9.8 Diffraction by a circular aperture, 292. 

9.9 Diffraction by small apertures, 297. 

9.10 Scattering by a conducting sphere at short wavelengths, 299. 
References and suggested reading, 304. 

Problems, 305. 

chapter 10. M agnetohydrodynamics and Plasma Physics 309 

10.1 Introduction and definitions, 309. 

10.2 Magnetohydrodynamic equations, 311. 

10.3 Magnetic diffusion, viscosity, and pressure, 313. 

10.4 Magnetohydrodynamic flow, 316. 

10.5 Pinch effect, 320. 

10.6 Dynamic model of the pinch effect, 322. 

10.7 Instabilities, 326. 

10.8 Magnetohydrodynamic waves, 329. 

10.9 High-frequency plasma oscillations, 335. 

10.10 Short-wavelength limit, Debye screening distance, 339. 
References and suggested reading, 343. 

Problems, 343. 

chapter 11. Special Theory of Relativity 347 

11.1 Historical background and key experiments, 347. 

11.2 Postulates of special relativity, Lorentz transformation, 352. 

11.3 FitzGerald-Lorentz contraction and time dilatation, 357. 

11.4 Addition of velocities, Doppler shift, 360. 

11.5 Thomas precession, 364. 

11.6 Proper time and light cone, 369. 

11.7 Lorentz transformations as orthogonal transformations, 371. 

11.8 4-vectors and tensors, 374. 



Contents xv 

11.9 Covariance of electrodynamics, 377. 

11.10 Transformation of electromagnetic fields, 380. 

11.11 Covariance of the force equation and the conservation laws, 383. 
References and suggested reading, 386. 

Problems, 387. 

chapter 12. Relativistic-Particle Kinematics and Dynamics 391 

12.1 Momentum and energy of a particle, 391. 

12.2 Kinematics of decay of an unstable particle, 394. 

12.3 Center of momentum transformation, 397. 

12.4 Transformation of momenta from the center of momentum frame 
to the laboratory, 400. 

12.5 Covariant Lorentz force equation, Lagrangian and Hamiltonian, 
404. 

12.6 Relativistic corrections to the Lagrangian for interacting charged 
particles, 409. 

12.7 Motion in a uniform, static, magnetic field, 411. 

12.8 Motion in combined uniform, static, electric and magnetic fields, 
412. 

12.9 Particle drifts in nonuniform magnetic fields, 415. 

12.10 Adiabatic invariance of flux through an orbit, 419. 

References and suggested reading, 424. 
Problems, 425. 

chapter 13. Collisions between Charged Particles, Energy Loss, 

and Scattering 429 

13.1 Energy transfer in a Coulomb collision, 430. 

13.2 Energy transfer to a harmonically bound charge, 434. 

13.3 Classical and quantum-mechanical energy loss, 438. 

13.4 Density effect in collision energy loss, 443. 

13.5 Energy loss in an electronic plasma, 450. 

13.6 Elastic scattering of fast particles by atoms, 451. 

13.7 Mean square angle of scattering, multiple scattering, 456. 

13.8 Electrical conductivity of a plasma, 459. 
References and suggested, reading, 462. 
Problems, 462. 

chapter 14. Radiation by Moving Charges 464 

14.1 Lienard-Wiechert potentials and fields, 464. 

14.2 Larmor's radiated power formula and its relativistic 
generalization, 468. 

14.3 Angular distribution of radiation, 472. 

14.4 Radiation by an extremely relativistic charged particle, 475. 



xvi Contents 

14.5 General angular and frequency distributions of radiation from 
accelerated charges, 477. 

14.6 Frequency spectrum from relativistic charged particle in an instan- 
taneously circular orbit, synchrotron radiation, 481. 

14.7 Thomson scattering, 488. 

14.8 Scattering by quasi-free charges, 491. 

14.9 Cherenkov radiation, 494. 
References and suggested reading, 499. 
Problems, 500. 

chapter 15. Bremsstrahlung, Method of Virtual Quanta, Radia- 
tive Beta Processes 505 

15.1 Radiation emitted during collisions, 506. 

15.2 Bremsstrahlung in nonrelativistic Coulomb collisions, 509. 

15.3 Relativistic bremsstrahlung, 513. 

15.4 Screening, relativistic radiative energy loss, 516. 

15.5 Weizsacker-Williams method of virtual quanta, 520. 

15.6 Bremsstrahlung as the scattering of virtual quanta, 525. 

15.7 Radiation emitted during beta decay, 526. 

15.8 Radiation emitted in orbital-electron capture, 528. 
References and suggested reading, 533. 
Problems, 534. 

chapter 16. Multipole Fields 538 

16.1 Scalar spherical waves, 538. 

16.2 Multipole expansion of electromagnetic fields, 543. 

16.3 Properties of multipole fields, energy and angular momentum of 
radiation, 546. 

16.4 Angular distributions, 550. 

16.5 Sources of multipole radiation, multipole moments, 553. 

16.6 Multipole radiation in atoms and nuclei, 557. 

16.7 Radiation from a linear, center-fed antenna, 562. 

16.8 Spherical expansion of a vector plane wave, 566. 

16.9 Scattering by a conducting sphere, 569. 

16.10 Boundary- value problems with multipole fields, 574. 
References and suggested reading, 574. 

Problems, 574. 

chapter 17. Radiation Damping, Self-Fields of a Particle, 
Scattering and Absorption of Radiation by a Bound 
System 578 

17.1 Introductory considerations, 578. 

17.2 Radiative reaction force, 581. 



Contents xvu 

17.3 Abraham-Lorentz evaluation of the self-force, 584. 

17.4 Difficulties with the Abraham-Lorentz model, 589. 

17.5 Lorentz transformation properties of the Abraham-Lorentz model, 
Poincare stresses, 590. 

17.6 Covariant definitions of self-energy and momentum, 594. 

17.7 Integrodifferential equation of motion, including damping, 597. 

17.8 Line breadth and level shift of an oscillator, 600. 

17.9 Scattering and absorption of radiation by an oscillator, 602. 
References and suggested reading, 607. 

Problems, 608. 

appendix. Units and Dimensions 611 

Bibliography 622 

Index 625 



Introduction to 
Electrostatics 



Although amber and lodestone were known by the ancient Greeks, 
electrodynamics developed as a quantitative subject in about 80 years. 
Coulomb's observations on the forces between charged bodies were made 
around 1785. About 50 years later, Faraday was studying the effects of 
currents and magnetic fields. By 1864, Maxwell had published his famous 
paper on a dynamical theory of the electromagnetic field. 

We will begin our discussion with the subject of electrostatics— problems 
involving time-independent electric fields. Much of the material will be 
covered rather rapidly because it is in the nature of a review. We will use 
electrostatics as a testing ground to develop and use mathematical tech- 
niques of general applicability. 



1.1 Coulomb's Law 

All of electrostatics stems from the quantitative statement of Coulomb's 
law concerning the force acting between charged bodies at rest with respect 
to each other. Coulomb (and, even earlier, Cavendish) showed experi- 
mentally that the force between two small charged bodies separated a 
distance large compared to nheir dimensions 

(1) varied directly as the magnitude of each charge, 

(2) varied inversely as the square of the distance between them, 

(3) was directed along the line joining the charges, 

(4) was attractive if the bodies were oppositely charged and repulsive 
if the bodies had the same type of charge. 

Furthermore it was shown experimentally that the total force produced 

1 



2 Classical Electrodynamics 

on one small charged body by a number of the other small charged bodies 
placed around it was the vector sum of the individual two-body forces of 
Coulomb. 



1.2 Electric Field 

Although the thing that eventually gets measured is a force, it is useful 
to introduce a concept one step removed from the forces, the concept of 
an electric field due to some array of charged bodies. At the moment, the 
electric field can be defined as the force per unit charge acting at a given 
point. It is a vector function of position, denoted by E. One must be 
careful in its definition, however. It is not necessarily the force that one 
would observe by placing one unit of charge on a pith ball and placing it 
in position. The reason is that one unit of charge (e.g., 100 strokes of cat's 
fur on an amber rod) may be so large that its presence alters appreciably 
the field configuration of the array. Consequently one must use a limiting 
process whereby the ratio of the force on the small test body to the charge 
on it is measured for smaller and smaller amounts of charge. Experi- 
mentally, this ratio and the direction of the force will become constant as 
the amount of test charge is made smaller and smaller. These limiting 
values of magnitude and direction define the magnitude and direction of the 
electric field E at the point in question. In symbols we may write 

F = qE (1.1) 

where F is the force, E the electric field, and q the charge. In this equation 
it is assumed that the charge q is located at a point, and the force and the 
electric field are evaluated at that point. 

Coulomb's law can be written down similarly. If F is the force on a 
point charge q x , located at x lf due to another point charge q 2 , located at 
x 2 , then Coulomb's law is 

F = fc<h<? 2 [ Xl ~ X2 ) 3 (1.2) 

|Xi ~ x 2 | 3 

Note that q x and q 2 are algebraic quantities which can be positive or 
negative. The constant of proportionality k depends on the system of units 
used. 

The electric field at the point x due to a point charge q x at the point x x 
can be obtained directly: 

E(x) = k qi ( * ~ Xl) (1.3) 

|x - x x | 3 

as indicated in Fig. 1.1. The constant k is determined by the unit of charge 



[Sect. 1.2] Introduction to Electrostatics 

P 




Fig. 1.1 

chosen. In electrostatic units (esu), unit charge is chosen as that charge 
which exerts a force of one dyne on an equal charge located one centimeter 
away. Thus, with cgs units, k = 1 and the unit of charge is called the 
"stat-coulomb." In the mks system, k = (4 7 re )- 1 , where e (= 8.854 x 
10~ 12 farad/meter) is the permittivity of free space. We will use esu.* 

The experimentally observed linear superposition of forces due to many 
charges means that we may write the electric field at x due to a system of 
point charges q it located at x„ i = 1, 2, . . . , n, as the vector sum: 

W -|*S^ (L4) 

If the charges are so small and so numerous that they can be described by 
a charge density p(x') [if Aq is the charge in a small volume Ax Ay Az at 
the point x', then Aq = p(x') Ax Ay Az], the sum is replaced by an 
integral: 

E(x) = L(x')fc^^' (1.5) 

j |x — x r 

where d 3 x' = dx' dy' dz' is a three-dimensional volume element at x'. 

At this point it is worth while to introduce the Dirac delta function. In one 
dimension, the delta function, written d{x - a), is a mathematically improper 
function having the properties: 

(1) d(x — a) = for x ^ a, and 

6(x - a) dx = 1 if the region of integration includes x = a, and is zero 

otherwise. 
The delta function can be given rigorous meaning as the limit of a peaked curve 
such as a Gaussian which becomes narrower and narrower, but higher and 
higher, in such a way that the area under the curve is always constant. L. 
Schwartz's theory of distributions is a comprehensive rigorous mathematical 
approach to delta functions and their manipulations.! 

* The question of units is discussed in detail in the Appendix. 

t A useful, rigorous account of the Dirac delta function is given by Lighthill. (Full 
references for items cited in the text or footnotes by author only will be found in the 
Bibliography.) 



(2) 



4 Classical Electrodynamics 

From the definitions above it is evident that, for an arbitrary function f(x), 

(3) f(x) d(x - a) dx = /(a), and 

(4) (f(x) d'(x -a)dx= -/'(a), 

where a prime denotes differentiation with respect to the argument. 

If the delta function has as argument a function f(x) of the independent 
variable x, it can be transformed according to the rule, 



(5) */<*»- 2 di 



dx l/ 



d{x - x { ) , 



where /(x) is assumed to have only simple zeros, located at x = a^. 

In more than one dimension, we merely take products of delta functions in 
each dimension. In three dimensions, for example, 

(6) (5(x - X) = d(x x - JT X ) d(x 2 - X 2 ) d(x 3 - X 3 ) 

is a function which vanishes everywhere except at x = X, and is such that 

,„ v f «., „ (I if A V contains x = X, 

(7) <5(x - X) tPx = 

Jav [0 if A V does not contain x = X. 

Note that a delta function has the dimensions of an inverse volume in whatever 
number of dimensions the space has. 

A discrete set of point charges can be described with a charge density by 
means of delta functions. For example, 

n 

P (x) = £ fr c5(x - x .) (1>6) 

i = l 

represents a distribution of n point charges q u located at the points x^. Substitu- 
tion of this charge density (1.6) into (1.5) and integration, using the properties of 
the delta function, yields the discrete sum (1.4). 



1.3 Gauss's Law 

The integral (1.5) is not the most suitable form for the evaluation of 

electric fields. There is another integral result, called Gauss's law, which 

is often more useful and which furthermore leads to a differential equation 

for E(x). To obtain Gauss's law we first consider a point charge q and a 

closed surface S, as shown in Fig. 1.2. Let r be the distance from the 

charge to a point on the surface, n be the outwardly directed unit normal 

to the surface at that point, da be an element of surface area. If the electric 

field E at the point on the surface due to the charge q makes an angle 

with the unit normal, then the normal component of E times the area 

element is : „^„ fl 

_, , cos a , , . _. 

E • n da = q — — da (1.7) 

r 2 

Since E is directed along the line from the surface element to the charge q, 



[Sect. 1.3] 



Introduction to Electrostatics 




q inside S 



E n 



q outside S 




Fig. 1.2 Gauss's law. The normal component of electric field is integrated over the 
closed surface S. If the charge is inside (outside) S, the total solid angle subtended at 
the charge by the inner side of the surface is 4n (zero). 

cos 6 da = r 2 dQ, where dQ. is the element of solid angle subtended by da 
at the position of the charge. Therefore 

E-nda = qdQ. 0-8) 

If we now integrate the normal component of E over the whole surface, it 
is easy to see that 

„ , (477A if q lies inside S n Q> . 

is ^ if q lies outside S 



6 Classical Electrodynamics 

This result is Gauss's law for a single point charge. For a discrete set of 
charges, it is immediately apparent that 

E.nda = 4«-2ft (1.10) 



where the sum is over only those charges inside the surface S. For a 
continuous charge density p(x), Gauss's law becomes: 

i> E • n da = 4tt\ p (x) d z x (l.H) 

where V is the volume enclosed by S. 

Equation (1 . 1 1) is one of the basic equations of electrostatics. Note that 
it depends upon 

(1) the inverse square law for the force between charges, 

(2) the central nature of the force, 

(3) the linear superposition of the effects of different charges. 
Clearly, then, Gauss's law holds for Newtonian gravitational force fields, 
with matter density replacing charge density. 

It is interesting to observe that before Coulomb's observations 
Cavendish, by what amounted to a direct application of Gauss's law, did 
an experiment with two concentric conducting spheres and deduced that 
the power law of the force was inverse nth power, where n = 2.00 ± 0.02. 
By a refinement of the technique, Maxwell showed that n = 2.0 ± 0.00005. 
(See Jeans, p. 37, or Maxwell, Vol. 1, p. 80.) 



1.4 Differential Form of Gauss's Law 

Gauss's law can be thought of as being an integral formulation of the 
law of electrostatics. We can obtain a differential form (i.e., a differential 
equation) by using the divergence theorem. The divergence theorem states 
that for any vector field A(x) defined within a volume V surrounded by 
the closed surface 5" the relation 



<t A • n da = V • A d 3 x 



holds between the volume integral of the divergence of A and the surface 
integral of the outwardly directed normal component of A. The equation 
in fact can be used as the definition of the divergence (see Stratton, p. 4). 
To apply the divergence theorem we consider the integral relation 
expressed in Gauss's theorem: 



<P E • n da = 4tt- p(x) d 3 z 

•>s Jv 



I 



[Sect. 1 .5] Introduction to Electrostatics 7 

Now the divergence theorem allows us to write this as : 

(V . E - 4t77>) d z x = (1.12) 

>v 

for an arbitrary volume V. We can, in the usual way, put the integrand 
equal to zero to obtain 

V-E = 4*7> (1.13) 

which is the differential form of Gauss's law of electrostatics. This 
equation can itself be used to solve problems in electrostatics. However, 
it is often simpler to deal with scalar rather than vector functions of position, 
and then to derive the vector quantities at the end if necessary (see below). 

1.5 Another Equation of Electrostatics and the Scalar Potential 

The single equation (1.13) is not enough to specify completely the three 
components of the electric field E(x). Perhaps some readers know that a 
vector field can be specified completely if its divergence and curl are given 
everywhere in space. Thus we look for an equation specifying curl E as a 
function of position. Such an equation, namely, 

V x E = (1.14) 

follows directly from our generalized Coulomb's law (1.5): 



E(x) == f P (x') ^ ^1 d 3 

J H 'lx-x'l 3 



The vector factor in the integrand, viewed as a function of x, is the negative 
gradient of the scalar l/|x — x'| : 



|x - x'l 3 \|x - x'i/ 



Since the gradient operation involves x, but not the integration variable x', 
it can be taken outside the integral sign. Then the field can be written 

E(x) == -V ( p(x ' } d z x' (1.15) 

J |x — x'| 

Since the curl of the gradient of any scalar function of position vanishes 
(V x Vy) = 0, for all xp), (1.14) follows immediately from (1.15). 

Note that V x E = depends on the central nature of the force 
between charges, and on the fact that the force is a function of relative 
distances only, but does not depend on the inverse square nature. 



Classical Electrodynamics 




Fig. 1.3 

In (1.15) the electric field (a vector) is derived from a scalar by the 
gradient operation. Since one function of position is easier to deal with 
than three, it is worth while concentrating on the scalar function and giving 
it a name. Consequently we define the "scalar potential" <D(x) by the 

ec l uation: E=-VO (1.16) 

Then (1.15) shows that the scalar potential is given in terms of the charge 
density by C ( '\ 

0(x)= p{X) d 3 x' (1.17) 

J |x - x'| 

where the integration is over all charges in the universe, and <2> is arbitrary 
to the extent that a constant can be added to the right side of (1.17). 

The scalar potential has a physical interpretation when we consider the 
work done on a test charge q in transporting it from one point 04) to 
another point (B) in the presence of an electric field E(x), as shown in Fig. 
1.3. The force acting on the charge at any point is 

F = qE 
so that the work done in moving the charge from A to B is 

W = - ¥-dl= -q) E-dl (1.18) 

The minus sign appears because we are calculating the work done on the 
charge against the action of the field. With definition (1.16) the work can 
be written » /. 

W = q\ VO • d\ = q\ d®= g[0 7j > - OJ (1.19) 

which shows that q<& can be interpreted as the potential energy of the test 
charge in the electrostatic field. 

From (1.18) and (1.19) it can be seen that the line integral of the electric 
field between two points is independent of the path and is the negative of 
the potential difference between the points: 

Vdl= -(O^-OJ (1.20) 



i: 



[Sect. 1.6] 



Introduction to Electrostatics 



This follows directly, of course, from definition (1.16). If the path is closed, 
the line integral is zero, 



E • d\ = 



(1.21) 



a result that can also be obtained directly from Coulomb's law. Then 
application of Stokes's theorem [if A(x) is a vector field, S is an open 
surface, and C is the closed curve bounding S, 



(fc A • d\ = 



(V x A) • n da 



where d\ is a line element of C, n is the normal to S, and the path C is 
traversed in a right-hand screw sense relative to n] leads immediately back 
to V x E = 0. 



1.6 Surface Distributions of Charges and Dipoles and Discontinuities 
in the Electric Field and Potential 

One of the common problems in electrostatics is the determination of 
electric field or potential due to a given surface distribution of charges. 
Gauss's law (1.11) allows us to write down a partial result directly. If a 
surface S, with a unit normal n, has a surface-charge density of <r(x) 
(measured in statcoulombs per square centimeter) and electric fields E x 
and E 2 on either side of the surface, as shown in Fig. 1 .4, then Gauss's law 
tells us immediately that 

(E 2 - E x ) • n = 4tto- (1.22) 

This does not determine E x and E 2 unless there are no other sources of 
field and the geometry and form a are especially simple. All that (1.22) 
says is that there is a discontinuity of 4tt<j in the normal component of 
electric field in crossing a surface with a surface-charge density a, the 
crossing being made from the "inner" to the "outer" side of the surface. 



Fig. 1.4 Discontinuity in the normal com- 
ponent of electric field across a surface layer 
of charge. 



*-E 2 




10 Classical Electrodynamics 

The tangential component of electric field can be shown to be continuous 
across a boundary surface by using (1.21) for the line integral of E around 
a closed path. It is only necessary to take a rectangular path with negligible 
ends and one side on either side of the boundary. 

A general result for the potential (and hence the field, by differentiation) 
at any point in space (not just at the surface) can be obtained from (1.17) 
by replacing p d z x by a da: 



Js 



<D(x)= r^hda' (1.23) 

>S X — X 



Another problem of interest is the potential due to a dipole-layer 
distribution on a surface S. A dipole layer can be imagined as being formed 
by letting the surface S have a surface-charge density <j(x) on it, and 
another surface S', lying close to S, have an equal and opposite surface- 
charge density on it at neighboring points, as shown in Fig. 1.5. The 
dipole-layer distribution of strength D(x) is formed by letting S' approach 
infinitesimally close to S while the surface-charge density o(x) becomes 
infinite in such a manner that the product of a(x) and the local separation 
d(x) of S and S' approaches the limit Z>(x) : 

lim a(x)d(x) = D(x) (1.24) 

The direction of the dipole moment of the layer is normal to the surface S 
and in the direction going from negative to positive charge. 

To find the potential due to a dipole layer we can consider a single dipole 
and then superpose a surface density of them, or we can obtain the same 
result by performing mathematically the limiting process described in words 
above on the surface-density expression (1.23). The first way is perhaps 
simpler, but the second gives useful practice in vector calculus. Con- 
sequently we proceed with the limiting process. With n, the unit normal to 




Fig. 1.5 Limiting process involved in 
creating a dipole layer. 



[Sect. 1.6] 



Introduction to Electrostatics 



11 




Fig. 1.6 Dipole-layer geometry. 

the surface S, directed away from S', as shown in Fig. 1.6, the potential 
due to the two close surfaces is 

«,) = f _•&!. da > _ f — e&o_ da . 

Js |x — x'| Js' |x — x' + nd\ 

For small d we can expand |x — x' + nJ| _1 . Consider the general 
expression |x + a| _1 , where |a| < |x|. Then we write 



1 



1 



|x + a| sjx 2 + a 2 + 2a • x 

x + 



a; \a;/ 



This is, of course, just a Taylor's series expansion in three dimensions. Tn 
this way we find that the potential becomes [upon taking the limit (1.24)]: 

O(x) = D(x')n • V ( — ) da' (1.25) 

■ Js \|x — x'|/ 

Equation (1.25) has a simple geometrical interpretation. We note that 

rr/j 1 \ j f cos da' ,„ 

n • V ( da = = — dQ. 

\|x - x'|/ |x - x'| 2 

where dQ, is the element of solid angle subtended at the observation point 
by the area element da, as indicated in Fig. 1 .7. Note that dQ, has a positive 
sign if is an acute angle, i.e., when the observation point views the "inner" 
side of the dipole layer. The potential can be written: 



(D(x) = - D(x') dQ 

J si 



(1.26) 



12 Classical Electrodynamics 







Fig. 1.7 The potential at P due to the 
dipole layer D on the area element da' is 
just the negative product of D and the 
solid angle element dQ, subtended by da' 
at P. 



For a constant surface-dipole-moment density D, the potential is just the 
product of the moment and the solid angle subtended at the observation 
point by the surface, regardless of its shape. 

There is a discontinuity in potential in crossing a double layer. This 
can be seen by letting the observation point come infinitesimally close to 
the double layer. The double layer is now imagined to consist of two 
parts, one being a small disc directly under the observation point. The 
disc is sufficiently small that it is sensibly flat and has constant surface- 
dipole-moment density D. Evidently the total potential can be obtained 
by linear superposition of the potential of the disc and that of the remain- 
der. From (1.26) it is clear that the potential of the disc alone has a 
discontinuity of AttD in crossing from the inner to the outer side, being 
—2ttD on the inner side and -\-2ttD on the outer. The potential of the 
remainder alone, with its hole where the disc fits in, is continuous across 
the plane of the hole. Consequently the total potential jump in crossing 
the surface is: 0,-^ = 4^ (1.27) 

This result is analogous to (1.22) for the discontinuity of electric field in 
crossing a surface-charge density. Equation (1.27) can be interpreted 
"physically" as a potential drop occurring "inside" the dipole layer, and 
can be calculated as the product of the field between the two layers of 
surface charge times the separation before the limit is taken. 



1.7 Poisson's and Laplace's Equations 

In Sections 1.4 and 1.5 it was shown that the behavior of an electro- 
static field can be described by the two differential equations : 

V-E = 4tt P (1.13) 

and V x E = (1.14) 

the latter equation being equivalent to the statement that E is the gradient 
of a scalar function, the scalar potential <I> : 

E= -VO (1.16) 



[Sect. 1 .7] Introduction to Electrostatics 13 

Equations (1.13) and (1.16) can be combined into one partial differential 
equation for the single function O(x) : 

V 2 = -4t7 P (1.28) 

This equation is called Poisson's equation. In regions of space where there 
is no charge density, the scalar potential satisfies Laplace's equation: 

V 2 <D = (1.29) 

We already have a solution for the scalar potential in expression (1.17): 

0(x) = M*2-dV (1.17) 

J |x — X | 

To verify that this does indeed satisfy Poisson's equation (1.28) we operate 
with the Laplacian on both sides : 



V 2 0> = V 2 f-£&2_ d z x' = f P (x')V 2 ( — ) d*x' (1. 

J |x — x' J \|x — x'l/ 



30) 



We must now calculate the value of V 2 (l/|x — x'|). It is convenient (and 
allowable) to translate the origin to x' and so consider V 2 (l/r), where r is 
the magnitude of x. By direct calculation we find that V 2 (l/r) = for 

\r/ r dr 2 \ rl r dr 2 

At r = 0, however, the expression is undefined. Hence we must use a 
limiting process. Since we anticipate something like a Dirac delta function, 
we integrate V 2 (l/r) over a small volume V containing the origin. Then we 
use the divergence theorem to obtain a surface integral: 



Js dr \ r 



r 2 dQ. = -4tt- 



It has now been established that V 2 (l/r) = for r ^ 0, and that its volume 
integral is — 4-n-. Consequently we can write the improper (but mathe- 
matically justifiable) equation, V 2 (l/r) = — 4tt«3(x), or, more generally, 



\|x - x'l/ 



4t7<5(x - x') (1.31) 



Having established the singular nature of the Laplacian of 1/r, we can 
now complete our check on (1.17) as a solution of Poisson's equation. 



14 Classical Electrodynamics 

Equation (1.30) becomes 

V 2 0> = f p(x')[-47r<5(x - x')] d*x' = -4tt P (x) 

verifying the correctness of our solution (1.17). 

1.8 Green's Theorem 

If electrostatic problems always involved localized discrete or continuous 
distributions of charge with no boundary surfaces, the general solution 
(1.17) would be the most convenient and straightforward solution to any 
problem. There would be no need of Poisson's or Laplace's equation. In 
actual fact, of course, many, if not most, of the problems of electrostatics 
involve finite regions of space, with or without charge inside, and with 
prescribed boundary conditions on the bounding surfaces. These boundary 
conditions may be simulated by an appropriate distribution of charges 
outside the region of interest (perhaps at infinity), but (1.17) becomes 
inconvenient as a means of calculating the potential, except in simple cases 
(e.g., method of images). 

To handle the boundary conditions it is necessary to develop some new 
mathematical tools, namely, the identities or theorems due to George 
Green (1824). These follow as simple applications of the divergence 
theorem. The divergence theorem : 



{ 



V • A d 3 x = Q> A • n da 

v Js 

applies to any vector field A defined in the volume V bounded by the closed 
surface S. Let A = <f>Vy>, where <f> and \p are arbitrary scalar fields. Now 

V . (<jHip) = <f>V 2 y + V<£ • Vy> (l .32) 

and 

«iV^-n=^^ (1.33) 

on 

where d/dn is the normal derivative at the surface S (directed outwards 
from inside the volume V). When (1.32) and (1.33) are substituted into 
the divergence theorem, there results Green's first identity: 

f UV> + V<£ • Vy>) d z x = <t <f> ^ da (1 .34) 

Jv Js on 

If we write down (1.34) again with </> and xp interchanged, and then subtract 
it from (1.34), the V<£ • Vy> terms cancel, and we obtain Green's second 



[Sect. 1.9] 



Introduction to Electrostatics 



identity or Green's theorem : 



I 



. dtp deb 

dn dnJ 



da 



15 



(1.35) 



Poisson's differential equation for the potential can be converted into an 
integral equation if we choose a particular yt, namely \jR = l/|x — x'|, 
where x is the observation point and x' is the integration variable. Further, 
we put $ = <J>, the scalar potential, and make use of V 2 = — Anp. From 
(1.31) we know that V 2 (l/rt) == -4tt6(x - x), so that (1.35) becomes 



— 4tt(^(x')6(x 



X') + ^ />(X') 

R 



d 3 x' = 



•js 



L dn'\R/ 



10® 

Rdri. 



da' 



If the point x lies within the volume V, we obtain: 

« x) _ f £&L> ** + -L 6, \L |$ _ 0. ±(l)] da' (..36) 
v ' J v R ATrJslRdn' dn'\R/J 

If x lies outside the surface S, the left-hand side of (1.36) is zero. [Note 
that this is consistent with the interpretation of the surface integral as being 
the potential due to a surface-charge density a = (l/47r)(d<I>/dfl') and a 
dipole layer D = — (1/4tt)0. The discontinuities in electric field and 
potential (1.22) and (1.27) across the surface then lead to zero field and 
zero potential outside the volume V.] 

Two remarks are in order about result (1 .36). First, if the surface S goes 
to infinity and the electric field on S falls off faster than R~\ then the 
surface integral vanishes and (1.36) reduces to the familiar result (1.17). 
Second, for a charge-free volume the potential anywhere inside the volume 
(a solution of Laplace's equation) is expressed in (1.36) in terms of the 
potential and its normal derivative only on the surface of the volume. This 
rather surprising result is not a solution to a boundary-value problem, but 
only an integral equation, since the specification of both $ and d<D/d« 
{Cauehy boundary conditions) is an overspecification of the problem. This 
will be discussed in detail in the next sections, where techniques yielding 
solutions for appropriate boundary conditions will be developed using 
Green's theorem (1.35). 



1.9 Uniqueness of the Solution with Dirichlet or Neumann Boundary 
Conditions 

The question arises as to what are the boundary conditions appropriate 
for Poisson's (or Laplace's) equation in order that a unique and well- 
behaved (i.e., physically reasonable) solution exist inside the bounded 



16 Classical Electrodynamics 

region. Physical experience leads us to believe that specification of the 
potential on a closed surface (e.g., a system of conductors held at different 
potentials) defines a unique potential problem. This is called a Dirichlet 
problem, or Dirichlet boundary conditions. Similarly it is plausible that 
specification of the electric field (normal derivative of the potential) every- 
where on the surface (corresponding to a given surface-charge density) 
also defines a unique problem. Specification of the normal derivative 
is known as the Neumann boundary condition. We now proceed to prove 
these expectations by means of Green's first identity (1.34). 

We want to show the uniqueness of the solution of Poisson's equation, 
V 2 <I> = — 4irp, inside a volume V subject to either Dirichlet or Neumann 
boundary conditions on the closed bounding surface S. We suppose, to 
the contrary, that there exist two solutions Q) 1 and <J> 2 satisfying the same 
boundary conditions. Let 

U=® 2 -<$> 1 (1.37) 

Then V 2 £/ = inside V, and U = or dU/dn = on S for Dirichlet and 
Neumann boundary conditions, respectively. From Green's first identity 
(1.34), with (f> = \p = U, we find 

(UV 2 U +VU-VU)d 3 x= <£ U — da (1.38) 

Jv Js dn 

With the specified properties of U, this reduces (for both types of boundary 
conditions) to : 



|Vl/| 2 d 3 x = 
Jv 



which implies VU=0. Consequently, inside V, U is constant. For 
Dirichlet boundary conditions, U = on 5* so that, inside V, <J\ = d> 2 and 
the solution is unique. Similarly, for Neumann boundary conditions, the 
solution is unique, apart from an unimportant arbitrary additive constant. 

From the right-hand side of (1 .38) it is clear that there is also a unique 
solution to a problem with mixed boundary conditions (i.e., Dirichlet over 
part of the surface S, and Neumann over the remaining part). 

It should be clear that a solution to Poisson's equation with both O and 
d<&/dn specified on a closed boundary (Cauchy boundary conditions) does 
not exist, since there are unique solutions for Dirichlet and Neumann 
conditions separately and these will in general not be consistent. The 
question of whether Cauchy boundary conditions on an open surface define 
a unique electrostatic problem requires more discussion than is warranted 
here. The reader may refer to Morse and Feshbach, Section 6.2, pp. 692- 
706, or to Sommerfeld, Partial Differential Equations in Physics, Chapter 



[Sect. 1.9] 



Introduction to Electrostatics 



17 



II, for a detailed discussion of these questions. Morse and Feshbach base 
their treatment on the replacement of the partial differential equation by 
appropriate difference equations which they then solve by an iterative 
procedure. On the other hand, Sommerfeld bases his discussion on the 
method of characteristics where possible. The result of these investigations 
on which boundary conditions are appropriate is summarized in the table 
below (based on one given in Morse and Feshbach), where different types 



Type of 
Boundary 
Condition 








Type of Equation 








Elliptic: 
(Poisson's eq.) 


Hyperbolic 
(wave eq.) 


Parabolic 
(heat-con- 
duction eq.) 


— 


Dirichlet 




Not enough 
Too much 

Not enough 
Too much 






Open surface 




Not enough 






Unique, stable 
solution in one 
direction 




Closed surface 




Unique, stable 
solution 






Too much 




Neumann 




Not enough 




Open surface 




Unique, stable 
solution in one 
direction 




Closed surface 




Unique, stable 
solution in 
general 






Too much 




Cauchy 














Open surface 


Unphysical 
results 




Unique, stable 
solution 


Too much 


Closed surface 




Too much 






Too much 




Too much 





A stable solution is one for which small changes in the boundary conditions 
cause appreciable changes in the solution only in the neighborhood of the 
boundary. 



of partial differential equations and different kinds of boundary conditions 
are listed. 

Study of the table shows that electrostatic problems are specified only 
by Dirichlet or Neumann boundary conditions on a closed surface (part 
or all of which may be at iniinity, of course). 



18 



Classical Electrodynamics 



1.10 Formal Solution of Electrostatic Boundary-Value Problem with 
Green's Function 



The solution of Poisson's or Laplace's equation in a finite volume V with 
either Dirichlet or Neumann boundary conditions on the bounding surface 
S can be obtained by means of Green's theorem (1.35) and so-called 
"Green's functions." 

In obtaining result (1.36)— not a solution— we chose the function y to 
be l/|x — x'|, it being the potential of a unit point charge, satisfying the 
equation: , 

v ' 2 tbi) = "^ (x " x,) (L31) 

The function l/|x — x'| is only one of a class of functions depending on the 
variables x and x', and called Green's functions, which satisfy (1.31). In 

V' 2 G(x, x') = -4ttS(x - x') (1.39) 

1 



general, 
where 



G(x, x') = 



X - X' 



+ F(x, x') 



(1.40) 



with the function F satisfying Laplace's equation inside the volume V: 



V 2 F(x, x') = 



(1.41) 



In facing the problem of satisfying the prescribed boundary conditions 
on <D or d$>/dn, we can find the key by considering result (1.36). As has 
been pointed out already, this is not a solution satisfying the correct type 
of boundary conditions because both <J> and dQ>/dn appear in the surface 
integral. It is at best an integral equation for <J>. With the generalized 
concept of a Green's function and its additional freedom [via the function 
F(x, x')], there arises the possibility that we can use Green's theorem with 
tp = G(x, x') and choose F(x, x') to eliminate one or the other of the two 
surface integrals, obtaining a result which involves only Dirichlet or 
Neumann boundary conditions. Of course, if the necessary <7(x, x') 
depended in detail on the exact form of the boundary conditions, the 
method would have little generality. As will be seen immediately, this is 
not required, and G(x, x') satisfies rather simple boundary conditions on S. 

With Green's theorem (1.35), <j> = O, rp = G(x, x'), and the specified 
properties of G (1.39), it is simple to obtain the generalization of (1.36): 



<D(x) = P (x')G(x, x') dV + — <k 

Jv 4n Js 



G(x, x ) — - - d>(x') v ' 
on on 



da' 



(1.42) 



[Sect. 1.10] Introduction to Electrostatics 19 

The freedom available in the definition of G (1.40) means that we can make 
the surface integral depend only on the chosen type of boundary con- 
ditions. Thus, for Dirichlet boundary conditions we demand : 

G D (x,x') = for x' on S (1.43) 

Then the first term in the surface integral in (1.42) vanishes and the 
solution is 

O(x) = f p(x')G D (x, x') d 3 x' - i- <£ O(x') ^2 da' (1.44) 

Jv 4tt Js on 

For Neumann boundary conditions we must be more careful. The 
obvious choice of boundary condition on G(x, x') seems to be 

^ (x, x') = for x' on S 
dn' 

since that makes the second term in the surface integral in (1.42) vanish, 
as desired. But an application of Gauss's theorem to (1.39) shows that 

— da' = —477 
sdn' 

Consequently the simplest allowable boundary condition on G N is 

^(x,x')=-— forx'onS (1.45) 

dn' K S 

where S is the total area of the boundary surface. Then the solution is 

O(x) = <<D>s + f p(x)G N (x, x') d*x' + ±£> <^G N da' (1.46) 

Jv 477 Js on 

where (O)^ is the average value of the potential over the whole surface. 
The customary Neumann problem is the so-called "exterior problem" in 
which the volume V is bounded by two surfaces, one closed and finite, the 
other at infinity. Then the surface area S is infinite; the boundary 
condition (1.45) becomes homogeneous; the average value {Q>) s vanishes. 

We note that the Green's functions satisfy simple boundary conditions 
(1.43) or (1.45) which do not depend on the detailed form of the Dirichlet 
(or Neumann) boundary values. Even so, it is often rather involved (if 
not impossible) to determine G(x, x') because of its dependence on the 
shape of the surface S. We will encounter such problems in Chapter 2 
and 3. 

The mathematical symmetry property G(x, x') = G(x', x) can be proved 
for the Green's functions satisfying the Dirichlet boundary condition 
(1.43) by means of Green's theorem with <j> = G(x, y) and xp = G(x\ y), 



2® Classical Electrodynamics 

where y is the integration variable. Since the Green's function, as a function 
of one of its variables, is a potential due to a unit point charge, this sym- 
metry merely represents the physical interchangeability of the source and 
the observation points. For Neumann boundary conditions the symmetry 
is not automatic, but can be imposed as a separate requirement. 

As a final, important remark we note the physical meaning of F(x, x')> 
It is a solution of Laplace's equation inside V and so represents the 
potential of a system of charges external to the volume V. It can be 
thought of as the potential due to an external distribution of charges so 
chosen as to satisfy the homogeneous boundary conditions of zero 
potential (or zero normal derivative) on the surface S when combined with 
the potential of a point charge at the source point x'. Since the potential 
at a point x on the surface due to the point charge depends on the position 
of the source point, the external distribution of charge F(x, x') must also 
depend on the "parameter" x'. From this point of view, we see that the 
method of images (to be discussed in Chapter 2) is a physical equivalent 
of the determination of the appropriate F(x, x') to satisfy the boundary 
conditions (1.43) or (1.45). For the Dirichlet problem with conductors, 
^(x, x') can also be interpreted as the potential due to the surface-charge 
distribution induced on the conductors by the presence of a point charge 
at the source point x'. 



1.11 Electrostatic Potential Energy and Energy Density 

In Section 1.5 it was shown that the product of the scalar potential and 
the charge of a point object could be interpreted as potential energy. More 
precisely, if a point charge q i is brought from infinity to a point x, in a 
region of localized electric fields described by the scalar potential $ (which 
vanishes at infinity), the work done on the charge (and hence its potential 
energy) is given by 

W i =q^>{x i ) (1.47) 

The potential can be viewed as produced by an array of (n — 1) charges 
VjU — 1,2, • ■ . ,n — 1) at positions x,-. Then 

n-l 



; = 1 l^j Xj- 

so that the potential energy of the charge q t is 



(1.48) 



W< 



-*§i7^i c- 49 ) 



[Sect. 1.11] Introduction to Electrostatics 21 

It is clear that the total potential energy of all the charges due to all the 
forces acting between them is: 

n 

"-2 2^ (1 - 50) 



i = l j<i ' X * X< 



as can be seen most easily by adding each charge in succession. A more 
symmetric form can be written by summing over / andy unrestricted, and 
then dividing by 2 : 

w= iyy_is L _ (1 .5i) 

It is understood that i = j terms (infinite "self-energy" terms) are omitted 
in the double sum. 

For a continuous charge distribution [or, in general, using the Dirac 
delta functions (1.6)] the potential energy takes the form: 

1 [[pV)PW d * xd * x > (1.52) 

2 J J |x-x'| 

Another expression, equivalent to (1.52), can be obtained by noting that 
one of the integrals in (1.52) is just the scalar potential (1.17). Therefore 

W=--- L(x)0(x)d 3 x (1.53) 

Equations (1.51), (1.52), and (1.53) express the electrostatic potential 
energy in terms of the positions of the charges and so emphasize the 
interactions between charges via Coulomb forces. An alternative, and 
very fruitful, approach is to emphasize the electric field and to interpret 
the energy as being stored in the electric field surrounding the charges. To 
obtain this latter form, we make use of Poisson's equation to eliminate the 
charge density from (1.53): 

W-.= — f(DV 2 <D d z x 

877 J 

Integration by parts leads to the result: 

W = J_ f lV <D| 2 d*x = - \ |E| 2 d 3 x (1.54) 

where the integration is over all space. In (1.54) all explicit reference to 
charges has gone, and the energy is expressed as an integral of the square 
of the electric field over all space. This leads naturally to the identification 
of the integrand as an energy density w : 

w = - IE! 2 (1-55) 



22 Classical Electrodynamics 

j'P 




O 
Fig. 1.8 

This expression for energy density is intuitively reasonable, since regions 
of high fields "must" contain considerable energy. 

There is perhaps one puzzling thing about (1.55). The energy density is 
positive definite. Consequently its volume integral is necessarily non- 
negative. This seems to contradict our impression from (1.51) that the 
potential energy of two charges of opposite sign is negative. The reason 
for this apparent contradiction is that (1.54) and (1.55) contain "self- 
energy" contributions to the energy density, whereas the double sum in 
(1.51) does not. To illustrate this, consider two point charges q x and q 2 
located at x x and x 2 , as in Fig. 1.8. The electric field at the point P with 
coordinate x is 

E _ gi(x - x t ) g 2 (x - x 2 ) 
|x - Xl | 3 |x - x 2 | 3 
so that the energy density (1.55) is 

w = 2l! + <?2 2 g iq2 (x - x 2 ) - (x - x 2 ) 

877|x- Xl | 4 8t7|x-x 2 | 4 4t7|x-x 1 | 3 |x-x 2 | 3 K } 
Clearly the first two terms are self-energy contributions. To show that the 
third term gives the proper result for the interaction potential energv_we 
integrate over all space : 

A change of integration variable to p = (x — x^/lxi — x 2 | yields 

HW-p^X-Lfj^ti^ (1.58) 

|x x - x 2 | Att J P 3 \p + n| 3 

where n is a unit vector in the direction (x x — x 2 ). By straightforward 
integration the dimensionless volume integral can be shown to have the 
value 4?r, so that the interaction energy reduces to the expected value. 

Forces acting between charged bodies can be obtained by calculating 
the change in the total electrostatic energy of the system under small 
virtual displacements. Examples of this are discussed in the problems. 
Care must be taken to exhibit the energy in a form showing clearly those 



[Probs. 1] Introduction to Electrostatics 23 

factors which vary with a change in configuration and those which are 
kept constant. 

As a simple illustration we calculate the force per unit area on the surface 
of a conductor with a surface-charge density <r(x). In the immediate 
neighborhood of the surface the energy density is 

w = — |E| 2 = 2t7(7 2 (1.59) 

If we now imagine a small outward displacement Ax of an elemental area 
Aa of the conducting surface, the electrostatic energy decreases by an 
amount which is the product of energy density w and the excluded volume 

AxAa: aw= -2tto 2 Aci Ax (1.60) 

This means that there is an outward force per unit area equal to Ino 2 = w 
at the surface of the conductor. This result is normally derived by taking 
the product of the surface-charge density and the electric field, with care 
taken to eliminate the electric field due to the element of surface-charge 
density itself. 

REFERENCES AND SUGGESTED READING 

On the mathematical side, the subject of delta functions is treated simply but rigor- 
ously by 

Lighthill. 
For a discussion of different types of partial differential equations and the appropriate 
boundary conditions for each type, see 

Morse and Feshbach, Chapter 6, 

Sommerfeld, Partial Differential Equations in Physics, Chapter II, 

Courant and Hilbert, Vol. II, Chapters III-VI. 
The general theory of Green's functions is treated in detail by 

Friedman, Chapter 3, 

Morse and Feshbach, Chapter 7. 
The general theory of electrostatics is discussed extensively in many of the older books. 
Notable, in spite of some old-fashioned notation, are 

Maxwell, Vol. 1, Chapters II and IV, 

Jeans, Chapters II, VI, VII. 
Of more recent books, mention may be made of the treatment of the general theory by 

Stratton, Chapter III, and parts of Chapter II. 



PROBLEMS 

1.1 Use Gauss's theorem to prove the following statements : 

(a) Any excess charge placed on a conductor must lie entirely on its 
surface. (A conductor by definition contains charges capable of moving 
freely under the action of applied electric fields.) 



24 Classical Electrodynamics 

(b) A closed, hollow conductor shields its interior from fields due to 
charges outside, but does not shield its exterior from the fields due to 
charges placed inside it. 

(c) The electric field at the surface of a conductor is normal to the surface 
and has a magnitude 4tto, where a is the charge density per unit area on the 
surface. 

1.2 Two infinite, conducting, plane sheets of uniform thicknesses t x and t 2 , 
respectively, are placed parallel to one another with their adjacent faces 
separated by a distance L. The first sheet has a total charge per unit area 
(sum of the surface-charge densities on either side) equal to q lt while the 
second has q 2 . Use symmetry arguments and Gauss's law to prove that 

(a) the surface-charge densities on the adjacent faces are equal and 
opposite; 

(b) the surface-charge densities on the outer faces of the two sheets are 
the same; 

(c) the magnitudes of the charge densities and the fields produced are 
independent of the thicknesses t x and t 2 and the separation L. 

Find the surface-charge densities and fields explicitly in terms of q x and 
q 2 , and apply your results to the special case q x = —q 2 = Q. 

1.3 Each of three charged spheres of radius a, one conducting, one having a 
uniform charge density within its volume, and one having a spherically 
symmetric charge density which varies radially as r n (n > —3), has a total 
charge Q. Use Gauss's theorem to obtain the electric fields both inside and 
outside each sphere. Sketch the behavior of the fields as a function of 
radius for the first two spheres, and for the third with n = —2, +2. 

1.4 The time-average potential of a neutral hydrogen atom is given by 



„, e~ aT I . a.r\ 

®=1 — I 1 + 2) 



where q is the magnitude of the electronic charge, and a -1 = a J 2. Find 
the distribution of charge (both continuous and discrete) which will give 
this potential and interpret your result physically. 

1.5 A simple capacitor is a device formed by two insulated conductors adjacent 
to each other. If equal and opposite charges are placed on the conductors, 
there will be a certain difference of potential between them. The ratio of 
the magnitude of the charge on one conductor to the magnitude of the 
potential difference is called the capacitance (in electrostatic units it is 
measured in centimeters). Using Gauss's law, calculate the capacitance of 

(a) two large, fiat, conducting sheets of area A, separated by a small 
distance d; 

(b) two concentric conducting spheres with radii a, b (b > a) ; 

(c) two concentric conducting cylinders of length L, large compared to 
their radii a, b {b > a). 

(d) What is the inner diameter of the outer conductor in an air-filled 
coaxial cable whose center conductor is B&S#20 gauge wire and whose 
capacitance is 0.5 micromicrofarad/cm ? 0.05 micromicrofarad/cm ? 

1.6 Two long, cylindrical conductors of radii a x and a 2 are parallel and 
separated by a distance d which is large compared with either radius. 



[Probs. 1] Introduction to Electrostatics 25 

Show that the capacitance per unit length is given approximately by 

where a is the geometrical mean of the two radii. 

Approximately what B&S gauge wire (state diameter in millimeters 
as well as gauge) would be necessary to make a two-wire transmission line 
with a capacitance of 0.1 ju/if/cm if the separation of the wires was 0.5 cm? 
1.5 cm? 5.0 cm? 

1.7 (a) For the three capacitor geometries in Problem 1.5 calculate the total 
electrostatic energy and express it alternatively in terms of the equal and 
opposite charges Q and — Q placed on the conductors and the potential 
difference between them. 

(b) Sketch the energy density of the electrostatic field in each case as a 
function of the appropriate linear coordinate. 

1.8 Calculate the attractive force between conductors in the parallel plate 
capacitor (Problem 1.5a) and the parallel cylinder capacitor (Problem 1.6) 
for 

(a) fixed charges on each conductor; 

(b) fixed potential difference between conductors. 

1.9 Prove the mean value theorem: For charge-free space the value of the 
electrostatic potential at any point is equal to the average of the potential 
over the surface of any sphere centered on that point. 

1.10 Use Gauss's theorem to prove that at the surface of a curved charged 
conductor the normal derivative of the electric field is given by 



\ BE _ / 1 
E~dn ~ 



— —\ 



where R ± and R 2 are the principal radii of curvature of the surface. 

1.11 Prove Green's reciprocation theorem: If O is the potential due to a volume- 
charge density p within a volume V and a surface-charge density a on the 
surface S bounding the volume V, while €>' is the potential due to another 
charge distribution p and <x', then 

p$' cPx + crO' da = p'O d 3 x + a'<& da 

Jv Js Jr Js 

1.12 Prove Thomson's theorem : If a number of conducting surfaces are fixed in 
position and a given total charge is placed on each surface, then the electro- 
static energy in the region bounded by the surfaces is a minimum when the 
charges are placed so that every surface is an equipotential. 

1.13 Prove the following theorem: If a number of conducting surfaces are 
fixed in position with a given total charge on each, the introduction of an 
uncharged, insulated conductor into the region bounded by the surfaces 
lowers the electrostatic energy. 



Boundary- Value Problems 
in Electrostatics: I 



Many problems in electrostatics involve boundary surfaces on which 
either the potential or the surface-charge density is specified. The formal 
solution of such problems was presented in Section 1.10, using the method 
of Green's functions. In practical situations (or even rather idealized 
approximations to practical situations) the discovery of the correct Green's 
function is sometimes easy and sometimes not. Consequently a number of 
approaches to electrostatic boundary-value problems have been developed, 
some of which are only remotely connected to the Green's function 
method. In this chapter we will examine two of these special techniques : 
(1) the method of images, which is closely related to the use of Green's 
functions; (2) expansion in orthogonal functions, an approach directly 
through the differential equation and rather remote from the direct 
construction of a Green's function. Other methods of attack, such as the 
use of conformal mapping in two-dimensional problems, will be omitted. 
For a discussion of conformal mapping the interested reader may refer to 
the references cited at the end of the chapter. 



2.1 Method of Images 

The method of images concerns itself with the problem of one or more 
point charges in the presence of boundary surfaces, e.g., conductors either 
grounded or held at fixed potentials. Under favorable conditions it is 
possible to infer from the geometry of the situation that a small number of 
suitably placed charges of appropriate magnitudes, external to the region 
of interest, can simulate the required boundary conditions. These charges 

26 



[Sect. 2.2] 



Boundary-Value Problems in Electrostatics: I 



27 



-f = 



U — $ = o 



-9 



-r 



Fig. 2.1 Solution by method of 
images. The original potential 
problem is on the left, the 
equivalent-image problem on 
the right. 



are called image charges, and the replacement of the actual problem with 
boundaries by an enlarged region with image charges but no boundaries is 
called the method of images. The image charges must be external to the 
volume of interest, since their potentials must be solutions of Laplace's 
equation inside the volume; the "particular integral" (i.e., solution of 
Poisson's equation) is provided by the sum of the potentials of the charges 
inside the volume. 

A simple example is a point charge located in front of an infinite plane 
conductor at zero potential, as shown in Fig. 2.1. It is clear that this is 
equivalent to the problem of the original charge and an equal and opposite 
charge located at the mirror- image point behind the plane defined by the 
position of the conductor. 



2.2 Point Charge in the Presence of a Grounded Conducting 
Sphere 



As an illustration of the method of images we consider the problem 
illustrated in Fig. 2.2 of a point charge q located at y relative to the origin 
around which is centered a grounded conducting sphere of radius a. * We 
seek the potential O(x) such that 0(|x| = a) = 0. By symmetry it is 
evident that the image charge q' (assuming that only one image is needed) 
will lie on the ray from the origin to the charge q. If we consider the charge 
q outside the sphere, the image position y' will lie inside the sphere. The 

* The term grounded is used to imply that the surface or object is held at the same 
potential as the point at infinity by means of some fine conducting connector. The 
connection is assumed not to disturb the potential distribution. But arbitrary amounts 
of charge of either sign can flow onto the object from infinity in order to maintain its 
potential at "ground" (usually taken to be zero potential). A conductor held at a fixed 
potential is essentially the same situation, except that a voltage source is interposed 
between the object and "ground." 



28 



Classical Electrodynamics 




Fig. 2.2 Conducting sphere of 

radius a, with charge q and image 

charge q'. 



potential due to the charges q and q' is : 
$(x) = 7-2—, + 



x - y| |x - y'| 



(2.1) 



We now must try to choose q' and |y'| such that this potential vanishes at 
|x| = a. If n is a unit vector in the direction x, and n' a unit vector in the 
direction y, then 



$(*) = ~ + 



xn — yn'\ \xn — y'n'\ 



(2.2) 



If x is factored out of the first term and y' out of the second, the potential 
at x = a becomes: 



(fr(x = a) = 



q 



n n 



+ 





q' 


y' 


a 
n n 

y' 



From the form of (2.3) it will be seen that the choices: 



(2.3) 



y _a_ 

a y' 



make 0(a; = a) = 0, for all possible values of n • n'. Hence the magnitude 
and position of the image charge are 



, a , a 

q = --q, y = — 
y y 



(2.4) 



[Sect. 2.2] 



Boundary- Value Problems in Electrostatics: 



29 



We note that, as the charge q is brought closer to the sphere, the image 
charge grows in magnitude and moves out from the center of the sphere. 
When q is just outside the surface of the sphere, the image charge is equal 
and opposite in magnitude and lies just beneath the surface. 

Now that the image charge has been found, we can return to the original 
problem of a charge q outside a grounded conducting sphere and consider 
various effects. The actual charge density induced on the surface of the 
sphere can be calculated from the normal derivative of <D at the surface: 



a = — 



j_ao> 

A-ndx 



A-ncf \y. 



(;)■ 



('-$) 



(2.5) 



( 1+ «_!_ 2 « cosr y 

\ y 2 y i 

where y is the angle between x and y. This charge density in units of 
— q/4TTa 2 is shown plotted in Fig. 2.3 as a function of y for two values of 
y\a. The concentration of charge in the direction of the point charge q is 
evident, especially for y\a — 2. It is easy to show by direct integration 
that the total induced charge on the sphere is equal to the magnitude of the 
image charge, as it must according to Gauss's law. 



Fig. 2.3 Surface-charge density a 
induced on the grounded sphere 
of radius a due to the presence 
of a point charge q located a dis- 
tance y away from the center of 
the sphere, a is plotted in units of 
—q/4TTa 2 as function of the angular 
position y away from the radius 
to the charge for y = 2a, 4a. 



4ira 2 <r 

-q 




30 Classical Electrodynamics 

^dF=2ir(T 2 da 



Fig. 2.4 



The force acting on the charge q can be calculated in different ways. 
One (the easiest) way is to write down immediately the force between the 
charge q and the image charge q' . The distance between them is y — y' = 
2/(1 — a 2 /y 2 ). Hence the attractive force, according to Coulomb's law, is: 

For large separations the force is an inverse cube law, but close to the 
sphere it is proportional to the inverse square of the distance away from 
the surface of the sphere. 

The alternative method for obtaining the force is to calculate the total 
force acting on the surface of the sphere. The force on each element of 
area da is lira 2 da, where a is given by (2.5), as indicated in Fig. 2.4. But 
from symmetry it is clear that only the component parallel to the radius 
vector from the center of the sphere to q contributes to the total force. 
Hence the total force acting on the sphere (equal and opposite to the force 
acting on q) is given by the integral: 

1 1 + — cos 7 

\ y* y J 

Integration immediately yields (2.6). 

The whole discussion has been based on the understanding that the 
point charge q is outside the sphere. Actually, the results apply equally for 
the charge q inside the sphere. The only change necessary is in the surface- 
charge density (2.5), where the normal derivative out of the conductor is 
now radially inwards, implying a change in sign. The reader may transcribe 
all the formulas, remembering that now y < a. The angular distributions 
of surface charge are similar to those of Fig. 2.3, but the total induced 
surface charge is evidently equal to — q, independent ofy. 



[Sect. 2.3] Boundary-Value Problems in Electrostatics: I 31 

2.3 Point Charge in the Presence of a Charged, Insulated, 
Conducting Sphere 

In the previous section we considered the problem of a point charge q 
near a grounded sphere and saw that a surface-charge density was induced 
on the sphere. This charge was of total amount q' = —aqjy, and was 
distributed over the surface in such a way as to be in equilibrium under all 
forces acting. 

If we wish to consider the problem of an insulated conducting sphere 
with total charge Q in the presence of a point charge q, we can build up 
the solution for the potential by linear superposition. In an operational 
sense, we can imagine that we start with the grounded conducting sphere 
(with its charge q' distributed over its surface). We then disconnect the 
ground wire and add to the sphere an amount of charge (Q — q'). This 
brings the total charge on the sphere up to Q. To find the potential we 
merely note that the added charge (Q — q') will distribute itself uniformly 
over the surface, since the electrostatic forces due to the point charge q are 
already balanced by the charge q'. Hence the potential due to the added 
charge (Q — q') will be the same as if a point charge of that magnitude 
were at the origin, at least for points outside the sphere. 

The potential is the superposition of (2.1) and the potential of a point 
charge (Q — q') at the origin: 



<D(x) = 



|x-y[ 



Q + a - q 

aq + y - (2.8) 



a 

y 2 



The force acting on the charge q can be written down directly from 
Coulomb's law. It is directed along the radius vector to q and has the 
magnitude : 

qa 3 (2y 2 — a 2 ) 



*"S 



Q 



y{y 2 - a 2 ) 2 J y 



(2.9) 



In the limit of y > a, the force reduces to the usual Coulomb's law for two 
small charged bodies. But close to the sphere the force is modified because 
of the induced charge distribution on the surface of the sphere. Figure 2.5 
shows the force as a function of distance for various ratios of Qjq. The 
force is expressed in units of q 2 /y 2 ; positive (negative) values correspond 
to a repulsion (attraction). If the sphere is charged oppositely to q, or is 



32 



Classical Electrodynamics 




Fig. 2.5 The force on a point charge q due to an insulated, conducting sphere of radius 
a carrying a total charge Q. Positive values mean a repulsion, negative an attraction. 
The asymptotic dependence of the force has been divided out. Fy 2 /q 2 is plotted versus 
yja for Q/q = -1, 0, 1, 3. Regardless of the value of Q, the force is always attractive 
at close distances because of the induced surface charge. 

uncharged, the force is attractive at all distances. Even if the charge Q is 
the same sign as q, however, the force becomes attractive at very close 
distances. In the limit of Q > q, the point of zero force (unstable equili- 
brium point) is very close to the sphere, namely, at y ~ a{\ + iVq/Q). 
Note that the asymptotic value of the force is attained as soon as the charge 
q is more than a few radii away from the sphere. 

This example exhibits a general property which explains why an excess 
of charge on the surface does not immediately leave the surface because of 
mutual repulsion of the individual charges. As soon as an element of 
charge is removed from the surface, the image force tends to attract it 
back. If sufficient work is done, of course, charge can be removed from 
the surface to infinity. The work function of a metal is in large part just 
the work done against the attractive image force in order to remove an 
electron from the surface. 



[Sect. 2.5] Boundary-Value Problems in Electrostatics: I 33 



2.4 Point Charge near a Conducting Sphere at Fixed Potential 

Another problem which can be discussed easily is that of a point charge 
near a conducting sphere held at a fixed potential V. The potential is the 
same as for the charged sphere, except that the charge (Q — q') at the 
center is replaced by a charge; (Va). This can be seen from (2.8), since at 
|x| = a the first two terms cancel and the last term will be equal to V as 
required. Thus the potential is 

$(x) = — 1 — ^- - 1 - — 



|x - y| 



a 
x y 

y 2 



+ rf ( 2 - 10 > 

x 



The force on the charge q due to the sphere at fixed potential is 

(2.11) 



F--2 



Va- qaf 



y*L (y*-a*fJy 

For corresponding values of Va/q and Q/q this force is very similar to that 
of the charged sphere, shown in Fig. 2.5, although the approach to 
the asymptotic value (Vaqfy 2 ) is more gradual. For Va > q, the unstable 
equilibrium point has the equivalent location y ~ a(\ + ^Vq/Va). 



2.5 Conducting Sphere in a Uniform Electric Field by Method 
of Images 

As a final example of the method of images we consider a conducting 
sphere of radius a in a uniform electric field E . A uniform field can be 
thought of as being produced by appropriate positive and negative charges 
at infinity. For example, if there are two charges ± Q, located at positions 
z = ^R, as shown in Fig. 2.6a, then in a region near the origin whose 
dimensions are very small compared to R there is an approximately 
constant electric field E ^ 2 Q/R 2 parallel to the z axis. In the limit as 
R, Q^-co, with Q/R 2 constant, this approximation becomes exact. 

If now a conducting sphere of radius a is placed at the origin, the 
potential will be that due to the charges ±Q at =fR and their images 
=FQalRatz= Ta 2 /R: 

Q Q 



o = 



( r 2 + #2 + lr R cos Of 4 (r 2 + R 2 - 2rR cos 0) H 
^Q + ^2 (2 12) 

R ( r z + ±- + ll2L cos e) R (r 2 + °L - ^ cos 6 
\ R 2 R J \ R 2 R 



34 



Classical Electrodynamics 





. — ■■ — 


P 








~~~~~— --- -__-« 


z=-R 







z = R 



(a) 




z=-R 



(b) 
Fig. 2.6 Conducting sphere in a uniform electric field by the method of images. 

where <J> has been expressed in terms of the spherical coordinates of the 
observation point. In the first two terms R is much larger than r by 
assumption. Hence we can expand the radicals after factoring out R 2 . 
Similarly, in the third and fourth terms, we can factor out r 2 and then 
expand. The result is : 



<D = 



2Q a , 2Qa 3 

-^ r cos V -\ *r — cos 

R 2 R 2 r 2 



+ 



(2.13) 



where the omitted terms vanish in the limit R —*- oo. In that limit 2Q/R 2 
becomes the applied uniform field, so that the potential is 



(2.14) 



O = —EJr — —\ cos 6 



The first term (— E z) is, of course, just the potential of a uniform field E 
which could have been written down directly instead of the first two terms 
in (2.12). The second is the potential due to the induced surface charge 
density or, equivalently, the image charges. Note that the image charges 
form a dipole of strength D = Qa/R x 2a 2 /R = E a s . The induced 
surface-charge density is 



477 dr 



= — E cos 

4tt 



(2.15) 



[Sect. 2.6] Boundary-Value Problems in Electrostatics: I 35 

We note that the surface integral of this charge density vanishes, so that 
there is no difference between a grounded and an insulated sphere. 



2.6 Method of Inversion 

The method of images for a sphere and related topics discussed in the 
previous sections suggest that there is some sort of equivalence of solutions 
of potential problems under the reciprocal radius transformation, 

a 2 
r->r' = - (2.16) 

r 

This equivalence forms the basis of the method of inversion, and trans- 
formation (2.16) is called inversion in a sphere. The radius of the sphere is 
called the radius of inversion, and the center of the sphere, the center of 
inversion. The mathematical equivalence is contained in the following 
theorem : 

Let 0(r, 6, <f>) be the potential due to a set of point charges q t at the 
points (r„ f , </> t ). Then the potential 

O'(r,0 ; «£) = ^O^-\0,<^ (2.17) 

is the potential due to charges, 

q/ = - qi (2.18) 

located at the points (a 2 /r 4 , d t , </>,). 

The proof of the theorem is as follows. The potential <b{r, d, <f>) can be 
written as 



*=2 



<it 



Vr 2 + r, 2 - 2rr t 



cosy* 



where y 4 is the angle between the radius vectors x and x t . Under trans- 
formation (2.16) the angles remain unchanged. Consequently the new 
potential <J>' is 






- , 2 2a 2 

— + r/ r t cos y t 



r r 



36 




Classical Electrodynamics 

$ . p 



-1.2q 



$ ' p 




-6q 



Fig. 2.7 



By factoring (r//r 2 ) out of the square root, this can be written 

(?) 



O 



'(r, 6, <f>) = ^ 



r + — - 2r — cos y, 



This proves the theorem. 

Figure 2.7 shows a simple configuration of charges before and after 
inversion. The potential O' at the point P due to the inverted distribution 
of charge is related by (2.17) to the original potential O at the point P' in 
the figure. 

The inversion theorem has been stated and proved with discrete charges. 
It is left as an exercise for the reader to show that, if the potential O 
satisfies Poisson's equation, 

V 2 0> = -Anp 



the new potential <D' (2.17) also satisfies Poisson's equation, 

V 2 <D'(r, d, $) = -47rp'(r, Q, cf>) 
where the new charge density is given by 



P '(r,d,cf>)=^j P (£,e,+) 



(2.19) 



(2.20) 



The connection between this transformation law for charge densities and 
the law (2.18) for point charges can be established by considering the 
charge density as a sum of delta functions : 

p(x) = 2,q i d(x-x i ) 



[Sect. 2.6] Boundary-Value Problems in Electrostatics: I 37 

In terms of spherical coordinates centered at the center of inversion the 
charge density can be written 

P {r, B, <f>) = y qi m - &d - 2 Kr ~ rd 

r r i 

where d(Q, — Q. t ) is the angular delta function whose integral over solid 
angle gives unity, and d(r — r t ) is the radial delta function.* Under 
inversion the angular factor is unchanged. Consequently we have 

p{j > 6 ><i>) = 2** Q - °*> f 2 6 {^ - r >) 

The radial delta function can be transformed according to rule 5 at the 
end of Section 1.2 as 

A-ti 



Then 



P (£,o,t) = 2^°-°^ 



6 /^,2\2 



and the inverted charge density (2.20) becomes 

P '(r, d, <f>) = ^ 2* (fj ^ x ~ x <') = 2«'^ x ~ x/) 

i * i 

where x/ = (a 2 /r,-, 0, <£) and q- = (a/r t )q i: , as required by (2.18). 

With the transformation laws for charges and volume-charge densities 
given by (2.18) and (2.20), it will not come as a great surprise that the 
transformation of surface-charge densities is according to 

o'(r,d,$)= ^0(^,6,^ (2.21) 

Before treating any examples of inversion there are one or two physical 
and geometrical points which need discussion. First, in regard to the 
physical points, if the original potential problem is one where there are 
conducting surfaces at fixed potentials, the inverted problem will not in 
general involve the inversions of those surfaces held at fixed potentials. 
This is evident from (2.17), where the factor a\r shows that even if <I> is 
constant on the original surface the potential O' on the inverted surface is 

* The factor r { ~ 2 multiplying the radial delta function is present to cancel out the r 2 
which appears in the volume element d 3 x = r 2 dr dCl. 



38 



Classical Electrodynamics 




Fig. 2.8 Geometry of inversion. 
Center of inversion is at 0. Radius 
of inversion is a. The inversion of 
the surface 5 is the surface S', and 



vice versa. 



not. The only exception occurs when <J> vanishes on some surface. Then 
O' also vanishes on the inverted surface. 

One might think that, since ^> is arbitrary to the extent of an additive 
constant, we could make any surface in the original problem have zero 
potential and so also be at zero potential in the inverted problem. This 
brings us to the second physical point. The inverted potentials corre- 
sponding to two potential problems differing only by an added constant 
potential <J> represent physically different charge configurations, namely, 
charge distributions which differ by a point charge a<& located at the center 
of inversion. This can be seen from (2.17), where a constant term <D in <1> 
is transformed into (aO /r). Consequently care must be taken in applying 
the method of inversion to remember that the mapping of the point at 
infinity into the origin may introduce point charges there. If these are not 
wanted, they must be separately removed by suitable linear superposition. 

The geometrical considerations involve only some elementary points 
which can be proved very simply. The notation is shown in Fig. 2.8. Let 
O be the center of inversion, and a the radius of inversion. The inter- 
section of the sphere of inversion and the plane of the paper is shown as 
the dotted circle. A surface S intersects the page with the curve AB. The 
inverted surface S', obtained by transformation (2.16), intersects the page 
in the curve A'B'. The following facts are stated without proof: 

(a) Angles of intersection are not altered by inversion. 

(b) An element of area da on the surface S is related to an element of 
area da' on the inverted surface S' by da/da' = r 2 /r' 2 . 

(c) The inverse of a sphere is always another sphere [perhaps of infinite 
radius; see (d)]. 

(d) The inverse of any plane is a sphere which passes through the center 
of inversion, and conversely. 

Figure 2.9 illustrates the possibilities involved in (c) and (d) when the 
center of inversion lies outside, on the surface of, or inside the sphere. 



[Sect. 2.6] 



Boundary- Value Problems in Electrostatics: I 



39 



As a very simple example of the solution of a potential problem by 
inversion we consider an isolated conducting sphere of radius R with a 
total charge Q on it. The potential has the constant value Q/R inside the 
sphere and falls off inversely with distance away from the center for points 
outside the sphere. By a suitable choice of center of inversion and 
associated parameters we can obtain the potential due to a point charge q 
a distance i/away from an infinite, grounded, conducting plane. Evidently, 
if the center of inversion O is chosen to lie on the surface of the sphere of 
radius R, the sphere will invert into a plane. This geometric situation is 
shown in Fig. 2.10. Furthermore, if we choose the arbitrary additive 
constant potential % to have the value — QjR, the sphere and its inversion, 
the plane, will be at zero potential, while a point charge —aQ/R will appear 
at the center of inversion. In order that we end up with a point charge q a 
distance d away from the plane it is necessary to choose the radius of 
inversion to be a = {2Rd)' A and the initial charge, Q = —(Rj2df*q. The 
surface-charge density induced on the plane can be found easily from (2.21). 
Since the charge density on the sphere is uniform over its surface, the 
induced charge density on the plane varies inversely as the cube of the 
distance away from the origin (as can be verified from the image solution; 
see Problem 2.1). 

If the center of inversion is chosen to lie outside the isolated uniformly 
charged sphere, it is clear from Fig. 2.9 that the inverted problem can be 




Fig. 2.9 Various possibilities for the inversion of a sphere. If the center of inversion O 

lies on the surface 5 of the sphere, the inverted surface S' is a plane; otherwise it is 

another sphere. The sphere of inversion is shown dotted. 



40 



Classical Electrodynamics 



■ — f— - 




2R 



■*! 



Fig. 2.10 Potential due to isolated, charged, 
conducting sphere of radius R is inverted to give 
the potential of a point charge a distance d 
away from an infinite, flat, conducting surface. 



made that of a point charge near a grounded conducting sphere, handled 
by images in Section 2.2. The explicit verification of this is left to Problem 
2.9. 

A very interesting use of inversion was made by Lord Kelvin in 1 847. 
He calculated the charge densities on the inner and outer surfaces of a thin, 
charged, conducting bowl made from a sphere with a cap cut out of it. The 
potential distribution which he inverted was that of a thin, flat, charged, 
circular disc (the charged disc is discussed in Section 3.12). As the shape 
of the bowl is varied from a shallow watch glass-like shape to an almost 
closed sphere, the charge densities go from those of the disc to those of a 
closed sphere, in the one limit being almost the same inside and out, but 
concentrated at the edges of the bowl, and in the other limit being almost 
zero on the inner surface and uniform over the outer surface. Numerical 
values are given in Kelvin's collected papers, p. 186, and in Jeans, pp. 
250-251. 



2.7 Green's Function for the Sphere; General Solution 
for the Potential 



In preceding sections the problem of a conducting sphere in the presence 
of a point charge has been discussed by the method of images. As was 
mentioned in Section 1.10, the potential due to a unit charge and its image 
(or images), chosen to satisfy homogeneous boundary conditions, is just 



[Sect. 2.7] 



Boundary-Value Problems in Electrostatics: 



41 



the Green's function (1.43 or 1.45) appropriate for Dirichlet or Neumann 
boundary conditions. Tn G^x, x') the variable x' refers to the location P' 
of the unit charge, while the variable x is the point P at which the potential 
is being evaluated. These coordinates and the sphere are shown in Fig. 
2. 1 1 . For Dirichlet boundary conditions on the sphere of radius a the 
potential due to a unit charge; and its image is given by (2.1) with q = 1 
and relations (2.4). Transforming variables appropriately, we obtain the 
Green's function: 

1 



G(x, x') = 

In terms of spherical coordinates this can be written : 

1 1 



G(x, x') = 




(2.22) 



(*■ + ** - 2^ cos yT l^ + a *_ lxx , cos y J 



(2.23) 



where y is the angle between x and x'. The symmetry in the variables x 
and x' is obvious in the form (2.23), as is the condition that G = if either x 
or x' is on the surface of the sphere. 




Fig. 2.11 



42 



Classical Electrodynamics 



For solution (1.44) of Poisson's equation we need not only G, but also 
dG/dri. Remembering that n' is the unit normal outwards from the 
volume of interest, i.e., inwards along x' toward the origin, we have 



dG 

dn' 



(x 2 - a 2 ) 



a{x 2 + a 2 — lax cos y)' 



(2.24) 



[Note that this is essentially the induced surface-charge density (2.5).] 
Hence the solution of Laplace's equation outside a sphere with the potential 
specified on its surface is, according to (1.44), 



0(x) = f U(a, B', <f>' 



a(x 2 — a 2 ) 



(x 2 + a 2 — 2a x cos yj 



7 dV (2.25) 



where dO.' is the element of solid angle at the point (a, 6', </>') and cos y = 
cos 6 cos 0' + sin sin 0' cos (<£ — <£')• For the interior problem, the 
normal derivative is radially outwards, so that the sign of dG/dn' is opposite 
to (2.24). This is equivalent to replacing the factor (x 2 — a 2 ) by (a 2 — x 2 ) 
in (2.25). For a problem with a charge distribution, we must add to (2.25) 
the appropriate integral in (1.44), with the Green's function (2.23). 



2.8 Conducting Sphere with Hemispheres at Different Potentials 

As an example of general solution for the potential outside a sphere 
with prescribed values of potential on its surface, we consider the con- 
ducting sphere of radius a made up of two hemispheres separated by a small 
insulating ring. The hemispheres are kept at different potentials. It will 
suffice to consider the potentials as ± V, since arbitrary potentials can be 
handled by superposition of the solution for a sphere at fixed potential 
over its whole surface. The insulating ring lies in the z = plane, as 
shown in Fig. 2.12, with the upper (lower) hemisphere at potential +V 
(-V). 




Fig. 2.12 



[Sect. 2.8] Boundary-Value Problems in Electrostatics: I 43 

From (2.25) the solution for 0(#, 6, <f>) is given by the integral: 

<D(*, 6,<f>) = T PVf \ \/(cos 0') - f ° d(cos 0')} , 2 "f~ a2) ^ 

47rJo Uo J-i ) (a + a — lax cos y)^ 

(2.26) 

By a suitable change of variables in the second integral (0' — >■ 7r — 0', 
<f>' -+ <f>' + 77), this can be cast in the form : 



*(*,M) = 



Fa(z 2 



Jo Jo 



d(cos d')[(a 2 + x 2 — laxcos y)~ 3/i 

- (a 2 + x 2 + lax cos y) - ^] (2.27) 

Because of the complicated dependence of cos y on the angles (0', </>') and 
(6, (f>), equation (2.27) cannot in general be integrated in closed form. 

As a special case we consider the potential on the positive z axis. Then 
cos y = cos 6' since = 0. The integration is elementary, and the 
potential can be shown to be 

(z 2 -«V 



<D(z) = V 



1 - 



sVz 2 + a 2 - 1 



(2.28) 



At 2 = a, this reduces to <D == F as required, while at large distances it 
goes asymptotically as ~ 3 Va 2 /2z 2 . 

In the absence of a closed expression for the integrals in (2.27), we can 
expand the denominator in power series and integrate term by term. 
Factoring out (a 2 + x 2 ) from each denominator, we obtain 



4tt(x 2 + 



2\ /*2jt /*1 

-4-U ^'U(cos0')[(l-2acos r )- % 
ar)* Jo Jo 

-(l + 2acosy)- M ] (2.29) 



where a = axj(a 2 + a 2 ). We observe that in the expansion of the radicals 
only odd powers of a cos y will appear: 



[(1 - 2a cos y)~ 3A — (1 + 2a cos y)~ 3A ] = 6a cos 7 + 35a 3 cos 3 y + 



(2.30) 



It is now necessary to integrate odd powers of cos y over d<f>' d(cos 0') : 

dcf>' d(cos 0') cos y = 7T cos 
Jo Jo 



Jo Jo 



d(cos 0') cos 3 y = - cos 0(3 - cos 2 0) 
4 



(2.31) 



44 Classical Electrodynamics 

If (2.30) and (2.31) are inserted into (2.29), the potential becomes 

°(*' *' *> = ^j l J cos Q 



2x £ \ (x* + ay 

(2.32) 



1+- ** f3-cos 2 6) + 
. 24(a 2 + x 2 ? K 



We note that only odd powers of cos 6 appear, as required by the symmetry 
of the problem. If the expansion parameter is (a 2 /x 2 ), rather than a 2 , the 
series takes on the form : 



o(M, <t>) = 3Va2 



2x 2 



n 7 « 2 P 3 fl 3 A , 

COS0 -cos 3 COS0) + 

12a; 2 \2 2 / 



(2.33) 



For large values of xja this expansion converges rapidly and so is a useful 
representation for the potential. Even for xja = 5, the second term in the 
series is only of the order of 2 per cent. It is easily verified that, for 
cos 0=1, expression (2.33) agrees with the expansion of (2.28) for the 
potential on the axis. [The particular choice of angular factors in (2.33) is 
dictated by the definitions of the Legendre polynomials. The two factors 
are, in fact, P x {cos 6) and ^(cos 6), and the expansion of the potential is 
one in Legendre polynomials of odd order. We shall establish this in a 
systematic fashion in Section 3.3.] 



2.9 Orthogonal Functions and Expansions 

The representation of solutions of potential problems (or any mathe- 
matical physics problem) by expansions in orthogonal functions forms a 
powerful technique that can be used in a large class of problems. The 
particular orthogonal set chosen depends on the symmetries or near 
symmetries involved. To recall the general properties of orthogonal 
functions and expansions in terms of them, we consider an interval (a, b) 
in a variable | with a set of real or complex functions U n (£), n = 1,2,..., 
orthogonal on the interval (a, b). The orthogonality condition on the 
functions UJJ) is expressed by 

V n *(0tf m (0 & = 0, m^n (2.34) 



r 



lfn = m, the integral is finite. We assume that the functions are normal- 
ized so that the integral is unity. Then the functions are said to be 
orthonormal, and they satisfy 



f 

*> a 



U n *(OU m (£)d£ = d nm (2.35) 



[Sect. 2.9] Boundary-Value Problems in Electrostatics: I 45 

An arbitrary function /(I), square integrable on the interval {a, b), can 
be expanded in a series of the orthonormal functions UJJ). If the number 
of terms in the series is finite (say N), 

f(£)^la n UM (2.36) 

n = l 

then we can ask for the "best" choice of coefficients a n so that we get the 
"best" representation of the function /(I). If "best" is defined as mini- 
mizing the mean square error M N : 



M N = P \m - I a n UM 

"a I n=l 



dt (2.37) 



it is easy to show that the coefficients are given by 






U n *(S)f(£) di (2.38) 

where the orthonormality condition (2.35) has been used. This is the 
standard result for the coefficients in an orthonormal function expansion. 
If the number of terms N in series (2.36) is taken larger and larger, we 
intuitively expect that our series representation of /(£) is "better" and 
"better." Our intuition will be correct provided the set of orthonormal 
functions is complete, completeness being defined by the requirement that 
there exist a finite number N such that for N > N the mean square error 
M N can be made smaller than any arbitrarily small positive quantity. Then 
the series representation 

oo 

Ia n U n (i)=f(0 (2.39) 

« = :l 

with a n given by (2.38) is said to converge in the mean to /(£). Physicists 
generally leave the difficult job of proving completeness of a given set of 
functions to the mathematicians. All orthonormal sets of functions 
normally occurring in mathematical physics have been proved to be 
complete. 

Series (2.39) can be rewritten with the explicit form (2.38) for the 
coefficients a n : 

/(£) = [ { I U n *(i')U n (i)} fin di' (2.40) 

Since this represents any function /(!) on the interval (a, b), it is clear that 
the sum of bilinear terms U n *(i-')U n {£) must exist only in the neighborhood 
of I' = |. In fact, it must be true that 

2 U n *(t')UM = <5(£' - |) (2.41) 



46 Classical Electrodynamics 

This is the so-called completeness or closure relation. It is analogous to the 
orthonormality condition (2.35), except that the roles of the continuous 
variable £ and the discrete index n have been interchanged. 

The most famous orthogonal functions are the sines and cosines, an 
expansion in terms of them being a Fourier series. If the interval in x is 
(—a/2, a/2), the orthonormal functions are 



2 . (iTrmxX fl (iTrmxX 

-sin I 1, /-cos I I 

a \ a / N a \ a 1 



where m is an integer. The series equivalent to (2.39) is customarily 
written in the form: . 

TO = 1 

where 

, 2 f a/2 ,, , (2irmx\ , 

A m = - f(x) cos dx 

a J -a/2 \ a I 

(2.43) 

B m = - f(x) sin I rfa; 

a ^-o/2 \ a / 

If the interval spanned by the orthonormal set has more than one 
dimension, formulas (2.34)-(2.39) have obvious generalizations. Suppose 
that the space is two dimensional, and that the variable I ranges over the 
interval (a, b) while the variable r\ has the interval (c, d). The orthonormal 
functions in each dimension are U n (tj) and V m (rj). Then the expansion of 
an arbitrary function /(I, rj) is 

/(*, V)=1I a n JJM)VJri) (2.44) 

n m 

where 

a nm = f " d£ T driU n *(i)V m \rj)f(S, rj) (2.45) 

If the interval (a, b) becomes infinite, the set of orthogonal functions 
U n (£) may become a continuum of functions, rather than a denumerable 
set. Then the Kronecker delta symbol in (2.35) becomes a Dirac delta 
function. An important example is the Fourier integral. Start with the 
orthonormal set of complex exponentials, 

UJx) = 4= e i(2 " mx/a) (2.46) 

m = 0, ±1, ±2, . . . , on the interval (—all, a/2), with the expansion: 

1 °o 

f(x) = -= y A m e i{ - 2lTmxla) (2.47) 

Ja *—' 

V m= — oo 



[Sect. 2.10] Boundary-Value Problems in Electrostatics: I 47 

where 

Am = _L j ° /2 e -n»™*Mf( x ')dx' (2.48) 

m ^/a J -a/2 

Then let the interval become infinite {a -> oo), at the same time trans- 
forming 



2-7rm 



k 



^ J-oo 27rJ-c 



dk 



-A(k) 



The resulting expansion, equivalent to (2.47), is 

1 



/(«)=! A{ky k *dk 

.. / Z7T J - oo 



where 



A(k) = -L | * e~ ikx f(x) dx 



(2.49) 



^277 

The orthogonality condition is 

— f°° e^ k - k ' }x dx = d(k - W) 

2.7T v — oo 

while the completeness relation is 

J_ * ^JK*-*') ^^ _ fi( x _ 3.') 

27T J-oo 

These last integrals serve as convenient representations of a delta function. 
We note in (2.50)-(2.53) the complete equivalence of the two continuous 
variables x and k. 



(2.50) 
(2.51) 

(2.52) 

(2.53) 



2.10 Separation of Variables; Laplace's Equation 
in Rectangular Coordinates 

The partial differential equations of mathematical physics are often 
solved conveniently by a method called separation of variables. In the 
process, one often generates orthogonal sets of functions which are useful 
in their own right. Equations involving the three-dimensional Laplacian 
operator are known to be separable in eleven different coordinate systems 



48 Classical Electrodynamics 

(see Morse and Feshbach, pp. 509, 655). We will discuss only three of these 
in any detail— rectangular, spherical, and cylindrical— and will begin with 
the simplest, rectangular coordinates. 
Laplace's equation in rectangular coordinates is 

a 2 o , a 2 o , a 2 o> n 

A solution of this partial differential equation can be found in terms of 
three ordinary differential equations, all of the same form, by the assumption 
that the potential can be represented by a product of three functions, one 
for each coordinate: 

<!>(*, y, z) = X(x) Y(y)Z(z) (2.55) 

Substitution into (2.54) and division of the result by (2.55) yields 

X(x) dx 2 + Y(y) dy 2 Z(z) dz 2 (2>56) 

where total derivatives have replaced partial derivatives, since each term 
involves a function of one variable only. If (2.56) is to hold for arbitrary 
values of the independent coordinates, each of the three terms must be 
separately constant: 

}_<£X = _ 2 
X dx 2 ~ 



Ydy* 
1 d 2 Z 



- = y 2, 

Zdz* r 

.2 



(2.57) 



where a 2 + /5 2 = y 2 

If we arbitrarily choose a 2 and /5 2 to be positive, then the solutions of the 
three or dinary d ifferential equations (2.57) are exp(±/aa;); exp(±/ / S«/) > 
exp (±Va 2 + £ 2 z). The potential (2.55) can thus be built up from the 
product solutions : 

At this stage a and @ are completely arbitrary. Consequently (2.58), by 
linear superposition, represents a very large class of solutions to Laplace's 
equation. 

To determine a and /S it is necessary to impose specific boundary 
conditions on the potential. As an example, consider a rectangular box, 
located as shown in Fig. 2.13, with dimensions (a, b, c) in the (x, y, z) 



[Sect. 2.10] Boundary-Value Problems in Electrostatics: I 

z 



49 



f = 0- 



Fig. 2.13 Hollow, rectangular 
box with five sides at zero 
potential, while the sixth (z = c) 
has the specified potential <D = 
V(x, y). 




-f = V(xy) 



4 4 



t. 



y = b 



f> = 



* = 



directions. All surfaces of the box are kept at zero potential, except the 
surface z = c, which is at a potential V(x, y). It is required to find the 
potential everywhere inside the box. Starting with the requirement that 
<E> = for x = 0, y = 0, z = 0, it is easy to see that the required forms of 
X, Y,Z are 

X = sin ax 



Y = sin /fy 

Z = sinh (Va 2 + 0*z) 



(2.59) 



In order that O = at x = a and y = b, it is necessary that cm = mr and 
(ib = rmr. With the definitions, 



mr 



OL„ = 



mv 
b 



Vnm =7r 



" 2 b 2 



(2.60) 



we can write the partial potential $ wm ; satisfying all the boundary 
conditions except one, 



<D nTO = sin (a w ar) sin ($„&) sinh (y nm z) 



(2.61) 



The potential can be expanded in terms of these 3>„ m with initially arbitrary 
coefficients (to be chosen to satisfy the final boundary condition): 

oo 

0(ic, y,z)= 2 A nm sin (a w *) sin (fi m y) sinh (y nm z) (2.62) 



50 Classical Electrodynamics 

There remains only the boundary condition <X> = V(x, y)atz = c: 

oo 

V(x, y) = 2 A nm sin (pt n x) sin (p m y) sinh (y WTO c) (2.63) 

n,m = l 

This is just a double Fourier series for the function V(x, y). Consequently 
the coefficients A nm are given by: 

4 f a P 

^»™ = ,. w : dx dyV(x, y) sin (a B a;) sin (0 m y) (2.64) 

flfc sinh (y nTO c) Jo Jo 

If the rectangular box has potentials different from zero on all six sides, 
the required solution for the potential inside the box can be obtained by a 
linear superposition of six solutions, one for each side, equivalent to (2.62) 
and (2.64). The problem of the solution of Poisson's equation, i.e., the 
potential inside the box with a charge distribution inside, as well as 
prescribed boundary conditions on the surface, requires the construction of 
the appropriate Green's function, according to (1.43) and (1.44). Discus- 
sion of this topic will be deferred until we have treated Laplace's equation 
in spherical and cylindrical coordinates. For the moment, we merely note 
that solution (2.62) and (2.64) is equivalent to the surface integral in the 
Green's function solution (1 .44). 



REFERENCES AND SUGGESTED READING 

Images and inversion are treated in many books ; among the better or more extensive 
discussions are those by 

Jeans, Chapter VIII, 

Maxwell, Vol. 1, Chapter XI, 

Smythe, Chapters IV and V. 
A truly encyclopedic source of examples with numerous diagrams is the book by 

Durand, especially Chapters III and IV. 
Durand discusses inversion on pp. 107-114. 

Conformal mapping techniques for the solution of two-dimensional potential problems 
are discussed by 

Durand, Chapter X, 

Jeans, Chapter VIII, Sections 306-337, 

Maxwell, Vol. 1, Chapter XII, 

Smythe, Chapter IV, Sections 4.09^1.29. 
There are, in addition, many engineering books devoted to the subject, e.g., 

Rothe, Ollendorff, and Polhausen. 
Elementary, but clear, discussions of the mathematical theory of Fourier series and 
integrals, and orthogonal expansions, can be found in 

Churchill, 

Hildebrand, Chapter 5. 
A somewhat old-fashioned treatment of Fourier series and integrals, but with many 
examples and problems, is given by 

Byerly. 



[Probs. 2] Boundary-Value Problems in Electrostatics: 



51 



PROBLEMS 

2.1 A point charge q is brought to a position a distance d away from an infinite 
plane conductor held at zero potential. Using the method of images, find: 

(a) the surface-charge density induced on the plane, and plot it; 

(b) the force between the plane and the charge by using Coulomb's law 
for the force between the charge and its image; 

(c) the total force acting on the plane by integrating 2-no 2 - over the whole 
plane; 

(d) the work necessary to remove the charge q from its position to 

infinity ; 

(e) the potential energy between the charge q and its image [compare the 
answer to id) and discuss]. 

(/) Find answer (d) in electron volts for an electron originally one 
angstrom from the surface. 

2.2 Using the method of images, discuss the problem of a point charge q 
inside a hollow, grounded, conducting sphere of inner radius a. Find 

(a) the potential inside the sphere; 

(b) the induced surface-charge density; 

(c) the magnitude and direction of the force acting on q. 

Is there any change in the solution if the sphere is kept at a fixed potential 
VI If the sphere has a total charge Q on it? 

2.3 Two infinite, grounded, conducting planes are located at x = a/2 and 
x = -a/2. A point charge q is placed between the planes at the point 
(x', y', z'), where —(a/2) < x' < (a/2). 

(a) Find the location and magnitude of all the image charges needed to 
satisfy the boundary conditions on the potential, and write down the 
Green's function G(x, x')- 

(b) If the charge a is at (*', 0, 0), find the surface-charge densities 
induced on each conducting plane and show that the sum of induced 
charge on the two planes is —q. 

2.4 Consider a potential problem in the half-space defined by z > 0, with 
Dirichlet boundary conditions on the plane z = (and at infinity). 

(a) Write down the appropriate Green's function G(x, x'). 

(b) If the potential on the plane z = is specified to be $ = V inside a 
circle of radius a centered at the origin, and <D = outside that circle, find 
an integral expression for the potential at the point P specified in terms of 
cylindrical coordinates (p, cj>, z). 

(c) Show that, along the axis of the circle (p = 0), the potential is given by 



$■ = V 



\ Va 2 + z 2 ) 



{d) Show that at large distances (p 2 + z 2 > a 2 ) the potential can be 
expanded in a power series in (p 2 + z 2 ) -1 , and that the leading terms are 






2 ( p 2 + z 2fA 



_ 3a 2 5(3p 2 a 2 + a 4 ) 

. 4(p 2 + z 2 ) + 8(p 2 + z 2 ) 2 



2.5 



2.6 



52 Classical Electrodynamics 

Verify that the results of (c) and (d) are consistent with each other in their 
common range of validity. 

An insulated spherical, conducting shell of radius a is in a uniform electric 

i?i £ ,i i 16 ^ 6r / is CUt int ° tw ° hemis pheres by a plane perpendicular 
to the held, find the force required to prevent the hemispheres from separa- 
te) if the shell is uncharged; 
(b) if the total charge on the shell is Q. 
A large parallel plate capacitor is made up of two plane conducting sheets 
one of which has a small hemispherical boss of radius a on its inner surface 
The conductor with the boss is kept at zero potential, and the other 
conductor is at a potential such that far from the boss the electric field 
between the plates is E . 

(a) Calculate the surface-charge densities at an arbitrary point dn the 
plane and on the boss, and sketch their behavior as a function of distance 
(or angle). 

(b) Show that the total charge on the boss has the magnitude 3£ a 2 /4 

(c) If, instead of the other conducting sheet at a different potential a 
point charge q is placed directly above the hemispherical boss at a distance 
d from its center, show that the charge induced on the boss is 

q — —q 1 ===== 

L dVd* + a 2 _ 

2.7 A line charge with linear charge density t is placed parallel to, and a distance 
R away from, the axis of a conducting cylinder of radius b held at fixed 
voltage such that the potential vanishes at infinity. Find 

(a) the magnitude and position of the image charge(s); 

(b) the potential at any point (expressed in polar coordinates with the 
line from the cylinder axis to the line charge as the x axis), including the 
asymptotic form far from the cylinder; 

(c) the induced surface-charge density, and plot it as a function of angle 
for Rib = 2, 4 in units of t/2tt-Z>; b 

(d) the force on the charge. 

2.8 (a) Find the Green's function for the two-dimensional potential problem 
with the potential specified on the surface of a cylinder of radius b, and 
show that the solution inside the cylinder is given by Poisson's integral: 

(b) Two halves of a long conducting cylinder of radius b are separated 
by a small gap, and are kept at different potentials V 1 and V 2 . Show that 
the potential inside is given by 



^t m v i + V 2 V, - V 2 
<£(r, 0) = -J— — ? + _1 ? tan" 



( 2br r\ 



where 6 is measured from a plane perpendicular to the plane through the 
g a P- 

(c) Calculate the surface-charge density on each half of the cylinder. 

(d) What modification is necessary in (a) if the potential is desired in the 
region of space bounded by the cylinder and infinity? 



[Probs. 2] Boundary-Value Problems in Electrostatics: I 53 

2.9 (a) An isolated conducting sphere is raised to a potential V. Write down 
the (trivial) solution for the electrostatic potential everywhere in space. 

(b) Apply the inversion theorem, choosing the center of inversion 
outside the conducting sphere. Show explicitly that the solution obtained 
for the potential is that of a grounded sphere in the presence of a point charge 
of magnitude - VR, where R is the inversion radius. 

(c) What is the physical situation described by the inverted solution if 
the center of inversion%es inside the conducting sphere? 

2.10 Knowing that the capacitance of a thin, flat, circular, conducting disc of 
radius a is {Ij^a and that; the surface-charge density on an isolated disc 
raised to a given potential is proportional to (a 2 - r 2 )~ 1/2 , where r is the 
distance from the center of the disc, 

(a) show that by inversion the potential can be found for the problem 
of an infinite, grounded, conducting plane with a circular hole in it and a 
point charge lying anywhere in the opening; 

(b) show that, for a unit point charge at the center of the opening, the 
induced charge density on the plane is 

<*(.r., 6,<t>) = , a 

^r 2 Vr 2 - a 2 

(c) show that (a) and (b) are a special case of the general problem, 
obtained by inversion of the disc, of a grounded, conducting, spherical 
bowl under the influence of a point charge located on the cap which is the 
complement of the bowl. 

2.11 A hollow cube has conducting walls defined by six planes x = y = z = 0, 
and x = y = z = a. The walls z = and z = a are held at a constant 
potential V. The other four sides are at zero potential. 

(a) Find the potential fl>(x, y, z) at any point inside the cube. 

(b) Evaluate the potential at the center of the cube numerically, accurate 
to three significant figures. How many terms in the series is it necessary to 
keep in order to attain this accuracy? Compare your numerical result 
with the average value of the potential on the walls. 

(c) Find the surface-charge density on the surface z = a. 



Boundary- Value Problems 
in Electrostatics: II 



In this chapter the discussion of boundary-value problems is con- 
tinued. Spherical and cylindrical geometries are first considered, and 
solutions of Laplace's equation are represented by expansions in series of 
the appropriate orthonormal functions. Only an outline is given of the 
solution of the various ordinary differential equations obtained from 
Laplace's equation by separation of variables, but an adequate summary of 
the properties of the different functions is presented. 

The problem of construction of Green's functions in terms of ortho- 
normal functions arises naturally in the attempt to solve Poisson's equation 
in the various geometries. Explicit examples of Green's functions are 
obtained and applied to specific problems, and the equivalence of the 
various approaches to potential problems is discussed. 



3.1 Laplace's Equation in Spherical Coordinates 

In spherical coordinates (r, 6, (j>), shown in Fig. 3.1, Laplace's equation 
can be written in the form : 

(KP) + — sin — l+— ^ = (3.1) 

rdr 2 r*smddd\ 30/ r 2 sin 2 6 d<j> 2 

If a product form for the potential is assumed, then it can be written: 

<D=^P(0)2(<£) (3.2) 

r 

54 



[Sect. 3.1] Boundary-Value Problems in Electrostatics: II 



55 




Fig. 3.1 

When this is substituted into (3.1), there results the equation 
UQ d 



* dr 2 r 2 sin dO 



sin — I 

\ del 



_UP_*Q = 
r 2 sin 2 6 d<j> 2 



If we multiply by r 2 sin 2 d/UPQ, we obtain 

1 



r sin 



1 d 2 U 
iU dr 2 r" sin OP dO 



d i . a dP 

sin u — 

dd 



+ ^^ = (3.3) 



The <f> dependence of the equation has now been isolated in the last term. 
Consequently that term must be a constant which we call (— m 2 ): 



This has solutions 



1 d 2 Q 2 

= — m 2 

Qd</> 2 



Q = e 



±im</> 



(3.4) 



(3.5) 



In order that Q be single valued, m must be an integer. By similar con- 
siderations we find separate equations for P(d) and U(r): 



1 



sin0d0\ 



d ( . dP\ 
— I sin u — ) 

id\ dd) 



/(/ + i) 



m 



P = 



*E - l(l + 1} u = o 

dr 2 r 2 



(3.6) 
(3.7) 



where /(/ + 1) is another real constant. 

From the form of the radial equation it is apparent that a single power 
of r (rather than a power series) will satisfy it. The solution is found to be : 

U = Ar^ 1 + Br~ l (3.8) 



but / is as yet undetermined. 



56 Classical Electrodynamics 



3.2 Legendre Equation and Legendre Polynomials 

The 6 equation for P(0) is customarily expressed in terms of x = cos 6, 
instead of 6 itself. Then it takes the form: 



d_ 

dx 



(«-*>t) + ( k, + 1) -[=*) p - (3 - 9) 

This equation is called the generalized Legendre equation, and its solutions 
are the associated Legendre functions. Before considering (3.9) we will 
outline the solution by power series of the ordinary Legendre differential 
equation with m 2 = : 

j- ((1 - x 2 ) f) + /(/ + 1)P = (3.10) 

The desired solution should be single valued, finite, and continuous on the 
interval — 1 < x < 1 in order that it represents a physical potential. The 
solution will be assumed to be represented by a power series of the form : 

P(x) = x*2 aj x j (3.11) 

i = 

where a is a parameter to be determined. When this is substituted into 
(3.10), there results the series: 

|{(«+/X«+j-iK*" +i " 2 

- [(a + ;)(<* + J + 1) ~ /(/ + l»i*" + '} = (3-12) 
In this expansion the coefficient of each power of a; must vanish separately. 
Fory = 0, 1 we find that 

if a =£ 0, then a(a — 1) = ] 



if a x ^ 0, then a(a + 1) = J 
while for a general j value 

■(a+jXa+j + 1) -Z(Z+1) 



a i + 2 — 



(3.13) 



(3.14) 



(a + j + l)(a +7 + 2) 

A moment's thought shows that the two relations (3.13) are equivalent and 
that it is sufficient to choose either a or a x different from zero, but not both. 
Making the former choice, we have a = or a = 1. From (3.14) we see 
that the power series has only even powers of x(<x. = 0) or only odd 
powers of x(<x. = 1). 



[Sect. 3.2] Boundary-Value Problems in Electrostatics: II 57 

For either of the series a = or a = 1 it is possible to prove the 
following properties: 

(a) the series converges for x % < 1, regardless of the value of /; 

(b) the series diverges at x ■— ±1, unless it terminates. 

Since we want a solution that is finite at x = ± 1, as well as for x % < 1, we 
demand that the series terminate. Since a andy are positive integers or 
zero, the recurrence relation (3.14) will terminate only if / is zero or a 
positive integer. Even then only one of the two series converges at x = ±1. 
If / is even (odd), then only the a = (a = 1) series terminates.* The 
polynomials in each case have x l as their highest power of x, the next 
highest being x l ~ 2 , and so on, down to x° (x) for / even (odd). By convention 
these polynomials are normalized to have the value unity at x = +1 and 
are called the Legendre polynomials of order /, P t (x). The first few 
Legendre polynomials are : 



P (x) = 1 

P^x) = X 

P 2 (x) = i(3x* - 1) 

p 3 (x) = \(5x* - 3x) 

p^x) = $(35a* - 30z 2 + 3) 



(3.15) 



By manipulation of the power series solutions (3.11) and (3.14) it is 
possible to obtain a compact representation of the Legendre polynomials, 
known as Rodriguez' formula: 

P t (x) = 4- — 7 (^-l) J (3-16) 

2 l l\dx l 

[This can be obtained by other, more elegant means, or by direct /-fold 
integration of the differential equation (3.10).] 

The Legendre polynomials form a complete orthogonal set of functions 
on the interval — 1 < x < 1. To prove the orthogonality we can appeal 
directly to the differential equation (3.10). We write down the differential 
equation for P z (x), multiply by P v (x), and then integrate over the interval: 



P Pi4*)\t(<\ - * 2 ) ~) + '(' + WO*) 

J— i \-dx\ ax' 



dx = (3.17) 



* For example, if / = the a == 1 series has a general coefficient a, = ajj + 1 for 
/ = 0, 2, 4 Thus the series is a (x + $x 3 + \x* + • • • .) This is just the power 



es at x = ± 1 . 



/l + x\ 
i expansion of a function Q^ix) — % In I I , which clearly diverg 

For each / value there is a similar function Qi(x) with logarithms in it as the partner to 
the well-behaved polynomial solution. See Magnus and Oberhettinger, p. 59. 



58 Classical Electrodynamics 

Integrating the first term by parts, we obtain 

I-i [ (X * " 1} S S + /(/ + W'C*)^*)] dx = ° ( 3 - 18 > 

If we now write down (3. 1 8) with / and /' interchanged and subtract it from 
(3.18), the result is the orthogonality condition: 

[/(/ + 1) - /'(/' + 1)] J' P v (x)Plx) dx = (3.19) 

For / ^ /', the integral must vanish. For / = /', the integral is finite. To 
determine its value it is necessary to use an explicit representation of the 
Legendre polynomials, e.g., Rodrigues' formula. Then the integral is 
explicitly: 

/ W <** = ^ j]£^ - v£v - v* 

Integration by parts / times yields the result: 

J> ( * )]2 dx = SS J> - *&* - v** 

The differentiation of (x 2 — l) 1 2/ times yields the constant (2/)!, so that 

jV*(*)] 2 dx = -2&L j\l - x y dx (3.20) 

The remaining integral is easily shown to be 2 2Z+1 (/!) 2 /(2/ + 1)! Con- 
sequently the orthogonality condition can be written : 



i 



P v {x)P t {x) dx = ^—^ d vi 



-i 21 + 1 

and the orthonormal functions in the sense of Section 2.9 are 



Ufc)= Jl^Pfc) 



(3.21) 



(3.22) 



Since the Legendre polynomials form a complete set of orthogonal 
functions, any function /(a;) on the interval — 1 < x < 1 can be expanded in 



+1 



-li 



! l 



-l 



Fig. 3.2 



[Sect. 3.2] Boundary-Value Problems in Electrostatics: II 

terms of them. The Legendre series representation is: 

oo 

f(x) = ^A l P l (x) 

1 = 

where 



A r = 



21 + 



! L 



f(x)P,(x) dx 



59 

(3.23) 
(3.24) 



As an example, consider the function shown in Fig. 3.2: 

f(x) = +1 for#>0 

= -1 forz<0 

Then 2l + 



/ + 1 r r 1 r° 

1Z_ Pi{x )dx-\ P t (x)dx 

2 LJo J-i J 



Since P t (x) is odd (even) about x = if / is odd (even), only the odd / 
coefficients are different from zero. Thus, for / odd, 

A l = (21 + 1) ( 1 P 1 (x) dx (3.25) 

Jo 

By means of Rodrigues' formula the integral can be evaluated, yielding 
,, = (-r" ,2 (2 '^^ 2)!! 0.26) 

where (2n + 1)!! = (2n + l)(2n - \)(2n -3)---x5x3xl. Thus the 
series for /(#) is: 

/(*) = fP^) - |P 3 (*) + HP 5 (*) ~ • ' ' (3.27) 

Certain recurrence relations among Legendre polynomials of different 
order are useful in evaluating integrals, generating higher-order poly- 
nomials from lower-order ones, etc. From Rodrigues' formula it is a 
straightforward matter to show that 

dP l+1 dP,_! 



- (21 + l)P t = 



(3.28) 



dx dx 

This result, combined with differential equation (3.10), can be made to 
yield various recurrence formulas, some of which are : 

(/ + 1)P I+1 - (21 + l)xP t + /P,-! = 



d Il±2 -x— l -(l + 1)P ? = 
dx dx 

dP 
dx 



(3.29) 



60 



Classical Electrodynamics 

As an illustration of the use of these recurrence formulas consider the 
evaluation of the integral: 



h=j_ *Pt 



(x)P t ix) dx 



(3.30) 



From the first of the recurrence formulas (3.29) we obtain an expression 
for xP(x). Therefore (3.30) becomes 

h = ~- { f^PfcW + 1)JW*) + WW*)] dx 

The orthogonality integral (3.21) can now be employed to show that the 
integral vanishes unless V = I ± 1, and that, for those values, 



/: 



2(1 + 1) 



xP z (x)P v (x) dx = « 



(21 + 1)(2/ + 3) 

2/ 
(21 - 1X2/ + 1) ' 



I' = 1 + 1 
V = 1-1 



(3.31) 



These are really the same result with the roles of / and /' interchanged. In 
a similar manner it is easy to show that 



x 2 Plx)P v (x) dx = 



2(1 + !)(/ + 2) 
(21 + 1X2/ + 3X2/ + 5) 

2(2/ 2 + 2/ - 1) 
(2/ - 1X2/ + 1X2/ + 3) 
where it is assumed that /' > /. 



r 



, V = 1 + 2 
" = I 



(3.32) 



3.3 Boundary-Value Problems with Azimuthal Symmetry 

From the form of the solution of Laplace's equation in spherical 
coordinates (3.2) it will be seen that, for a problem possessing azimuthal 
symmetry, m = in (3.5). This means that the general solution for such 
a problem is : 



®{r, Q)=2 [A t r l + Bf-WjP^cos 6) 



1 = 



(3.33) 



The coefficients A l and B x can be determined from the boundary condi- 
tions. Suppose that the potential is specified to be V(d) on the surface of a 
sphere of radius a, and it is required to find the potential inside the sphere. 
If there are no charges at the origin, the potential must be finite there. 
Consequently B t = for all /. The coefficients A h are found by evaluating 



[Sect. 3.3] Boundary-Value Problems in Electrostatics: II 61 

(3.33) on the surface of the sphere : 

V(d) = | Aja l Pfco& 0) (3.34) 

1 = 

This is just a Legendre series of the form (3.23), so that the coefficients At 
are: 

21 + 



A,= 



2a l Jo 



- I V(0)P,(cos 0) sin dd (3.35) 

Jo 



If, for example, V(6) is that of Section 2.8, with two hemispheres at equal 
and opposite potentials, 



V(G) = 



+ V, O<0<- 

2 (3.36) 

-V, -<0<tt 
2 



then the coefficients are proportional to those in (3.27). Thus the potential 
inside the sphere is : 



0(r, 6)=V 



r a PM -l(3 PM + U$ p * icose) - 



(3.37) 



To find the potential outside the sphere we merely replace (r/a) 1 by {ajr) l+1 . 
The resulting potential can be seen to be the same as (2.33), obtained by 
another means. 

Series (3.33), with its coefficients determined by the boundary conditions, 
is a unique expansion of the potential. This uniqueness provides a means 
of obtaining the solution of potential problems from a knowledge of the 
potential in a limited domain, namely on the symmetry axis. On the 
symmetry axis (3.33) becomes (with z = r): 

0>(z = r) = f [A z r l + B*r- (l+1) ] (3.38) 

1 = 

valid for positive z. For negative z each term must be multiplied by (— 1)'. 
Suppose that, by some means, we can evaluate the potential 0(z) at an 
arbitrary point z on the symmetry axis. If this potential function can be 
expanded in a power series in z = r of the form (3.38), with known 
coefficients, then the solution for the potential at any point in space is 
obtained by multiplying each power of r l and r~ (I+1> by ^(cos 0). 



62 



Classical Electrodynamics 




At the risk of boring the reader we return to the problem of the hemi- 
spheres at equal and opposite potentials. We have already obtained the 
series solution in two different ways, (2.33) and (3.37). The method just 
stated gives a third way. For a point on the axis we have found the closed 
form (2.28): 



<D(z = r )=V 



1 - 



r" — a 



rjr 2 + a 2 -l 



This can be expanded in powers of a 2 /r c ' 



* (z = r) = JL J ( _ iri «/ - IW - 1) (« 

V j = i 



J 



fio 



(3.39) 



Comparison with expansion (3.38) shows that only odd / values 
(/ = 2/ — 1) enter. The solution, valid for all points outside the sphere, 
is consequently: 

<D(r, 6) = -£ f (- 1)'"* (2j ~ VTV ~ *> (2f P^Ccos 0) (3.40) 

This is the same solution as already obtained, (2.33) and (3.37). 

An important expansion is that of the potential at x due to a unit point 
charge at x' : 



X — X 



T n r> 



(3.41) 



1 = 



where r< (/->) is the smaller (larger) of |x| and |x'|, and y is the angle 
between x and x', as shown in Fig. 3.3. This can be proved by rotating 
axes so that x' lies along the z axis. Then the potential satisfies Laplace's 
equation, possesses azimuthal symmetry, and can be expanded according 
to (3.33), except at the point x = x': 

00 

— = 2 <- A ** + B i r ~ il+1) ) p i(cos y) (3.42) 



x — x 



1 = 



[Sect. 3.3] 



Boundary-Value Problems in Electrostatics: II 



63 



If the point x is on the z axis, the right-hand side reduces to (3.38), while 
the left-hand side becomes : 



1 



1 



1 



x — x 



(r 2 + r' 2 - 2rr' cos y) A \r - r'\ 



Expanding (3.43), we find 



- x'l r^ 4i \rJ 



(3.43) 



(3.44) 



For points off the axis it is only necessary, according to (3.33) and (3.38), 
to multiply each term in (3.44) by i>j(cos y). This proves the general result 
(3.41). 

Another example is the potential due to a total charge q uniformly 
distributed around a circular ring of radius a, located as shown in Fig. 3.4, 
with its axis the z axis and its center at z = b. The potential at a point P 
on the axis of symmetry with z = r is just q divided by the distance AP: 



(J>(z = r) == 



(r 2 + c 2 — 2cr cos a) 



M 



(3.45) 



where c 2 = a 2 + b 2 and a = tan -1 (a/b). The inverse distance AP can be 
expanded using (3.41). Thus, for r > c, 



00 l 

®(z=r)=q^ J l ri P l (cos<x) 



(3.46) 



Fig. 3.4 Ring of charge of radius a and total 

charge q located on the z axis with center at 

z = b. 




64 Classical Electrodynamics 

For r < c, the corresponding form is : 



<D(z = r) = <?2 TTx ^( cos <*) ( 3 -47) 



The potential at any /ra/nf in space is now obtained by multiplying each 
member of these series by P{(cos 6) : 



00 , 

<D(r, 0) = q ]? ^1 P i( cos a ) P *( cos ) ( 3 - 48 ) 



—. r i+ 
i=o r > 

where r < (r > ) is the smaller (larger) of r and c. 



3.4 Associated Legendre Polynomials and the Spherical Harmonics 

So far we have dealt with potential problems possessing azimuthal 
symmetry with solutions of the form (3.33). These involve only ordinary 
Legendre polynomials. The general potential problem can, however, have 
azimuthal variations so that m ^ in (3.5) and (3.9). Then we need the 
generalization of ^(cos 6), namely, the solution of (3.9) with / and m both 
arbitrary. In essentially the same manner as for the ordinary Legendre 
functions it can be shown that in order to have finite solutions on the 
interval — 1 < x < 1 the parameter / must be zero or a positive integer and 
that the integer m can take on only the values — /, — (/ — 1), . . . , 0, . . . , 
(/ — 1), /. The solution having these properties is called an associated 
Legendre function P^ix). For positive m it is defined by the formula*: 

d m 
Pi m (x) = (- l) w (l - * 2 r /2 ~^ Pfc) (3-49) 

If Rodrigues' formula is used to represent P t (x), a definition valid for both 
positive and negative m is obtained : 

p^(x) = { —^- (1 - x 2 ) m/2 ^—— (x 2 - l) 1 (3.50) 

i v j 2l/ , v dx l+m . 

* The choice of phase for Pi m (x) is that of Magnus and Oberhettinger, and of E. U. 
Condon and G. H. Shortley in Theory of Atomic Spectra, Cambridge University Press 
(1953). For explicit expressions and recursion formulas, see Magnus and Oberhettinger, 
p. 54. 



[Sect. 3.4] Boundary-Value Problems in Electrostatics: II 65 

Pr m (x) and P, m (aO are proportional, since differential equation (3.9) 
depends only on m 2 and m is an integer. It can be shown that 

Pf "(*) = (~D m TTT^T, JVC*) ( 3 - 51 > 

(/ + m)\ 

For fixed m the functions P™(x) form an orthogonal set in the index / 
on the interval — 1 < x < 1 . By the same means as for the Legendre 
functions the orthogonality relation can be obtained: 



i. 



1 2 (I 4- mV 

P,,™(x)P, m (x) dx = —^— yi ^ mh b vi (3.52) 

' 2/ + l(/-m)! 



The solution of Laplace's equation was decomposed into a product of 
factors for the three variables r, 6, and <f>. It is convenient to combine the 
angular factors and construct orthonormal functions over the unit sphere. 
We will call these functions spherical harmonics, although this terminology 
is often reserved for solutions of the generalized Legendre equation (3.9). 
Our spherical harmonics are sometimes called "tesseral harmonics" in 
older books. The functions Q m (4d = eim * form a complete set of ortho- 
gonal functions in the index m on the interval < <f> <: 2tt. The functions 
Pj m (cos 6) form a similar set in the index / for each m value on the interval 
— 1 < cos 6 < 1. Therefore their product P™Q m will form a complete 
orthogonal set on the surface of the unit sphere in the two indices /, m. 
From the normalization condition (3.52) it is clear that the suitably 
normalized functions, denoted by Y lm {6, <f>), are : 

Y lm (e, & = Jl 1 -^ ( /~ m) '; pacc-s ey+ (3.53) 

v 477 (/ + m)! 
From (3.51) it can be seen that 

Vi»^) = (-1)X(M) (3-54) 

The normalization and orthogonality conditions are 



Jo Jo 



sin 6 d6 Y* m ,(d, <f>)Y lm (0, <f>) = d r fi m . m (3.55) 



The completeness relation, equivalent to (2.41), is 

00 I 

I I Y*Jd',<t>')Y lm {e,<i>) = d{<f>-4>')d(co^B-cose') (3.56) 

1=0 m= —I 

For a few small / values and m > the table shows the explicit form of 
the Y lm (6, <f>). For negative m values (3.54) can be used. 



66 



/ = 



/ = 1 



Classical Electrodynamics 
Spherical harmonics Y lm (d, <j>) 

Y -J- 

V4tt 



Jl sin 0g<* 



Y„ = 



r in = /_ cos o 



/ = 2 



y_ 2 =- 111 sin 2 0<? 2i * 



4V2r 
K, = - /!£ sin0 cos 0e'* 



r 2 „= /l/rCOS 2 0-i 



/ = 3 



4t7\2 



r 33 = - - /M sin 3 0e** 
4 V 4tt 



1 /105 . 



sin* cos 



y = - i /il sin 0(5 cos 2 - l)e<* 

4 V 4tt 

Yon = fl.(- cos 3 6 - - cos 



4tt \2 



I - cos 3 — - COS ) 
\2 2 J 



Note that, for m = 0, 



Y l0 (d,<f>) = J^^-P l ( C os&) 



(3.57) 



An arbitrary function g(6, </>) can be expanded in spherical harmonics : 

(3.58) 



00 I 

g(o, <t>) = I 2 i4 lm r Im (0, 0) 

1 = m — —I 



where the coefficients are 



A lm =jdQ Y t *(e, 4>)g{Q, <f>) 



[Sect. 3.5] Boundary-Value Problems in Electrostatics: II 



67 



A point of interest to us in the next section is the form of the expansion 
for = 0. With definition (3.57), we find: 



oo 



21 + 1 



where 



-m 



dQP l (cosd)g(6, ( f>) 



(3.59) 



(3.60) 



All terms in the series with m ^ vanish at 6 = 0. 

The general solution for a boundary-value problem in spherical coordi- 
nates can be written in terms of spherical harmonics and powers of r in a 
generalization of (3.33): 



<D(r,0,<£)=2 J [A lm r l + B lm r-« +1 ^Y lm (d, <f>) (3.61) 

1 = m ■= —I 

If the potential is specified on a spherical surface, the coefficients can be 
determined by evaluating (3.61) on the surface and using (3.58). 



3.5 Addition Theorem for Spherical Harmonics 

A mathematical result of considerable interest and use is called the 
addition theorem for spherical harmonics. Two coordinate vectors x and 
x', with spherical coordinates (r, 6, <f>) and (r', 6', <f>'), respectively, have an 
angle y between them, as shown in Fig. 3.5. The addition theorem 
expresses a Legendre polynomial of order / in the angle y in terms of 




Fig. 3.5 



68 Classical Electrodynamics 

products of the spherical harmonics of the angles 6, cf> and 6', cf>' : 

+ 



i 
Piioos y) = ^-j^ J Y <*( '' WimiO, <f>) (3.62) 



where cos y = cos 6 cos 0' + sin 6 sin 0' cos (<£ — <f>'). To prove this 
theorem we consider the vector x' as fixed in space. Then Pj(cos y) is a 
function of the angles 0, cf>, with the angles 6', $' as parameters. It may be 
expanded in a series (3.58): 

P,(cosy) = i J A v JB',<f>')Y v Jd,<i>) (3.63) 

l'=0 m= —V 

Comparison with (3.62) shows that only terms with /' = / appear. To see 
why this is so, note that, if coordinate axes are chosen so that x' is on the z 
axis, then y becomes the usual polar angle and Pj(cos y) satisfies the 
equation: 

V' 2 P,(cos y) + 1 1L±J± Pi(cos y) = o (3.64) 

r* 

where V' 2 is the Laplacian referred to these new axes. If the axes are now 
rotated to the position shown in Fig. 3.5, V' 2 = V 2 and r is unchanged.* 
Consequently Pj(cos y) still satisfies an equation of the form (3.64); i.e., 
it is a spherical harmonic of order /. This means that it is a linear com- 
bination of 7j TO 's of that order only: 

P l (cosy)= i A m id',<j>')Y lm {B,<f>) (3-65) 

m= —I 

The coefficients A m (6', <f>') are given by: 

ajo', </>') = \y* m {e, cf>)p l ( C os y) da (3.66) 

To evaluate this coefficient we note that it may be viewed, according to 
(3.60), as the m' = coefficient in an expansion of the function 
V4tt/(21 + 1) Yfijd, <f>) in a series of Y lmf {y, ft) referred to the primed 
axis of (3.64). From (3.59) it is then found that, since only one / value is 
present, coefficient (3.66) is 

AJQ', V) = ^^ [Y* m (0(y, ft, <i>{y, 0))], =o (3.67) 

In the limit y — > 0, the angles (0, <f>), as functions of (y, /?), go over into 

* The proof that V' 2 = V 2 under rotations follows most easily from noting that 
V 2 y» = V • Vy) is an operator scalar product, and that all scalar products are invariant 
under rotations. 



[Sect. 3.6] Boundary-Value Problems in Electrostatics: II 69 

(6', <f>'). Thus addition theorem (3.62) is proved. Sometimes the theorem 
is written in terms of P t m (co& 6) rather than Y lm . Then it has the form: 

Pj(cos y) = P ? (cos 0)Pj(cos 0') 

i 

+ 2 S (/ ~ m)! P ™(cos 6)P l m (cos 0') cos [m(<£ - <f>' )] (3.68) 
It the angle y goes to zero, there results a "sum rule" for the squares of 

Z irimC^r-^r 11 < 3 - 69) 

*-' , Air 

m= — I 

The addition theorem can be used to put expansion (3.41) of the potential 
at x due to a unit charge at x' into its most general form. Substituting 
(3.62) for Pj(cos y) into (3.41) we obtain 

ir-^Ti = ""2 2 jTTT 5^i Y '*- (e: ^^ « (3 - 70) 

l X ~ X I 1=0 m=-l Zl + lr > 

Equation (3.70) gives the potential in a completely factorized form in the 
coordinates x and x'. This is useful in any integrations over charge 
densities, etc., where one variable is the variable of integration and the 
other is the coordinate of the observation point. The price paid is that 
there is a double sum involved, rather than a single term. 



3.6 Laplace's Equation in Cylindrical Coordinates; Bessel Functions 

In cylindrical coordinates (p, <f>, z), as shown in Fig. 3.6, Laplace's 
equation takes the form : 

d P 2 pdp p 2 dcf> 2 dz 2 v ' 

The separation of variables is accomplished by the substitution : 

0( P , <M = P0>)fi(#Z( Z ) (3-72) 

In the usual way this leads to the three ordinary differential equations : 



dp 2 p dp 



d 2 7 

— - k 2 Z = 

dz 2 


(3.73) 


<^ + v 2 Q = 
dj> 2 * 


(3.74) 


fc 2 --]W = o 


(3.75) 



70 



Classical Electrodynamics 




Fig. 3.6 

The solutions of the first two equations are elementary : 

Z(z) = e ±kz 
(2(0) = e ±iv * 



(3.76) 



In order that the potential be single valued, v must be an integer. But 
barring some boundary-condition requirement in the z direction, the 
parameter k is arbitrary. For the present we will assume that k is real. 
The radial equation can be put in a standard form by the change of 
variable x = kp. Then it becomes 



'* + ii« + (i_r, R _o 



dx 2 



dx 



(3.77) 



This is Bessel's equation, and the solutions are called Bessel functions of 
order v. If a power series solution of the form: 



R(x) = x a ^ ajx j 

3=0 

is assumed, then it is found that 



and 



a = ±v 
1 



«2/ = - 



4/0' + ^ 



«2j-2 



(3.78) 
(3.79) 
(3.80) 



for j = 0, 1, 2, 3, ... . All odd powers of x j have vanishing coefficients. 
The recursion formula can be iterated to obtain 



«2i = 



(-lffla+1) 
2 2 V!T0- + a+l) 



(3.81) 



[Sect. 3.6] Boundary-Value Problems in Electrostatics: II 71 

It is conventional to choose the constant a = [2 x F(y. + l)] -1 . Then the 
two solutions are 

^)=(-lS /-^ (f 0.82) 

\2/^;!ro + v + i)\2/ 

These solutions are called Bessel functions of the first kind of order ±v. 
The series converge for all finite values of x. If v is not an integer, these 
two solutions J ±v (x) form a pair of linearly independent solutions to the 
second-order Bessel's equation. However, if v is an integer, it is well known 
that the solutions are linearly dependent. In fact, for v = m, an integer, 
it can be seen from the series representation that 

/-*(*) = (-D m / m (*) (3.84) 

Consequently it is necessary to find another linearly independent solution 
when m is an integer. It is customary, even if v is not an integer, to replace 
the pair J ±v (x) by J v (%) and N v (x), the Neumann function (or Bessel's 
function of the second kind) : 

N v (x) = •^cosvir-J.^) (3 g5) 

sin vn 

For v not an integer, N v (%) is clearly linearly independent of J v (x). In the 
limit v -► integer, it can be shown that N 9 (x) is still linearly independent 
of J v (x). As expected, it involves log a;. Its series representation is given 
in the reference books. 

The Bessel functions of the third kind, called Hankel functions, are 
defined as linear combinations of /„(x) and N v (x): 

H?\x) = J v (x) + iN v (*)] 

I (3-86) 

H™(x) = JXx)-iN v (x)j 

The Hankel functions form a fundamental set of solutions to Bessel's 
equation, just as do J v (x) and N v (x). 

The functions J v , N v , H ( }\ H™ all satisfy the recursion formulas: 

Q. v - 1 {x) + n v+1 0) = — Q v O) (3.87) 

x 

n v ^(x) - n v+1 (x) = 2 ?%^ (3<88) 

ax 



72 



Classical Electrodynamics 



where Q- V ( x ) is an y one °f the cylinder functions of order v. These may be 
verified directly from the series representation (3.82). 

For reference purposes, the limiting forms of the various kinds of 
Bessel functions will be given for small and large values of their argument. 
Only the leading terms will be given for simplicity : 



x < 1 J v ( x ) —> 



1 



N v ( x ) 



.5772 



I .- 



TT \X/ 

In these formulas v is assumed to be real and nonnegative. 

^ -. 7 / X /^ ( VTT Tt\~ 

cc>1, v J v (a;)->- / — cos I a; 1 

N ttx \ 2 4/ 

• 

N v (x) -> /— sin I x - — - -I 

N ttx \ 2 4/ J 



(3.89) 



(3.90) 



(3.91) 



The transition from the small x behavior to the large x asymptotic form 
occurs in the region of x ~ v. 

From the asymptotic forms (3.91) it is clear that each Bessel function 
has an infinite number of roots. We will be chiefly concerned with the 

roots of J v { x ) '• 

J v (xJ = 0, n = 1, 2, 3, . . . (3.92) 

x vn is the nth root of J£x). For the first few integer values of v, the first 
three roots are: 

v = 0, x 0n = 2.405, 5.520, 8.654 



v= 1, 



= 3.832,7.016, 10.173,... 



v = 2, x 2n = 5.136, 8.417, 11.620, . . . 
For higher roots, the asymptotic formula 

x vn ^ riTT + (y - i) " 

gives adequate accuracy (to at least three figures). Tables of roots are 
given in Jahnke and Emde, pp. 166-168. 

Having found the solution of the radial part of Laplace's equation in 
terms of Bessel functions, we can now ask in what sense the Bessel 
functions form an orthogonal, complete set of functions. We will consider 



[Sect. 3.6] Boundary-Value Problems in Electrostatics: II 73 

only Bessel functions of the first kind, and will show that Vp J v (x vn p/a), for 
fixed v > 0, n = 1, 2, . . . , form an orthogonal set on the interval < 
p < a. The demonstration starts with the differential equation satisfied by 
J (x old): 

If we multiply the equation by p/ v (# vn ,p/a) and integrate from to a, we 
obtain 






+ 

Integration by parts, combined with the vanishing of (pJJ v ') at p = 
(for v > 0) and p = a, leads to the result : 



^0 i/p ^0 



^P dp 



+ 



ff^-JW^fW^f)"'- 



If we now write down the same expression, with n and ri interchanged, 
and subtract, we obtain the orthogonality condition : 

(fin ~ x2 vn')j Q Pjy^vn' £ j^v(*v« ") dp = (3.94) 

By means of the recursion formulas (3.87) and (3.88) and the differential 
equation, the normalization integral can be found to be : 

1 pJv \ vn ' a) Jv V vn al dp = ^ [ J v + i(*v„)] 2<5 «'n ( 3 - 9 5) 

Assuming that the set of Bessel functions is complete, we can expand an 
arbitrary function of p on the interval ^ p < a in a Bessel-Fourier 
series : 

/(p) = ^A vn J v (x vn £) (3.96) 



74 
where 



Classical Electrodynamics 



A ™ = 2T2 2 , , fVoW^) dp (3.97) 

Our derivation of (3.96) involved the restriction v > 0. Actually it can 
be proved to hold for all v > — 1 . 

Expansion (3.96) and (3.97) is the conventional Fourier-Bessel series 
and is particularly appropriate to functions which vanish at p = a (e.g., 
homogeneous Dirichlet boundary conditions on a cylinder; see the 
following section). But it will be noted that an alternative expansion is 

possible in a series of functions \^~pJ v (ymPl a ) where y vn is the nth root of 
the equation [dJ v {x)]jdx = 0. The reason is that, in proving the ortho- 
gonality of the functions, all that is demanded is that the quantity 
[pJ v (Xp)(d/dp)J v (X' p)] vanish at the end points p = and p = a. The 
requirement is met by either % = x v Ja or X = y^/a, where J v ( x vr) = an( ^ 
J v '(y V n) — 0. The expansion in terms of the set V pJ v {y vn p\a) is especially 
useful for functions with vanishing slope at p = a. (See Problem 3.8.) 
A Fourier-Bessel series is only one type of expansion involving Bessel 



functions. Neumann series 



2 a Jv 

n=-0 



S z ) 



Kapteyn series 



rc=0 



Jv+n(( v + "» 



, and Schlomilch series 



2 aj v {nx) 



Ln = l 



are some of the other 



possibilities. The reader may refer to Watson, Chapters XVI-XIX, for a 
detailed discussion of the properties of these series. Kapteyn series occur 
in the discussion of the Kepler motion of planets and of radiation by 
rapidly moving charges (see Problems 14.7 and 14.8). 

Before leaving the properties of Bessel functions it should be noted that 
if, in the separation of Laplace's equation, the separation constant k 2 in 
(3.73) had been taken as —k 2 , then Z(z) would have been sin kz or cos kz 
and the equation for R(p) would have been : 



d 2 R ldR 

dp 2 p dp 



/c 2 + - 2 |K = 



With kp = x, this becomes 

dx 2 x dx \ x 2 / 



(3.98) 



(3.99) 



The solutions of this equation are called modified Bessel functions. It is 
evident that they are just Bessel functions of pure imaginary argument. 



[Sect. 3.7] Boundary-Value Problems in Electrostatics: II 



75 



The usual choices of linearly independent solutions are denoted by I v (x) 
and K v (x). They are defined by 



I£x) = r v J v (ix) 



K v (x) = - i^H^Xix) 



(3.100) 
(3.101) 



and are real functions for real x. Their limiting forms for small and large 
x are, assuming real v > : 



x<l J v (;r)^--1 h) 

~(v + 1)\2/ 



*v(*) 



-(lng) + 0.5772-..), 
L 2 \z/' 



1 



x > 1, v I v (x) -> —L= e a 

yJllTX 



K v (x) 



— e ' 
2x 



1 + 
1 + 






v = 



j-^0 



(3.102) 



(3.103) 



(3.104) 



3.7 Boundary-Value Problems in Cylindrical Coordinates 

The solution of Laplace's equation in cylindrical coordinates is $ = 
R(p)Q(<f>)Z(z), where the separate factors are given in the previous section. 
Consider now the specific boundary- value problem shown in Fig. 3.7. 
The cylinder has a radius a and a height L, the top and bottom surfaces 
being at z = L and z = 0. The potential on the side and the bottom of 
the cylinder is zero, while the top has a potential O = V(p, <f>). We want 
to find the potential at any point inside the cylinder. In order that O be 
single valued and vanish at z ■■= 0, 



Q(<f>) = A sin m<f> + B cos m<j> 
Z(z) = sinh kz 



(3.105) 



where v = m is an integer and k is a constant to be determined. The radial 
factor is 



R(p) = CJJkp) + DN m (kp) 



(3.106) 



16 



Classical Electrodynamics 

X 



# = o- 



■f =V(p,<t>) 




* = 



Fig. 3.7 



If the potential is finite at p = 0, D = 0. The requirement that the 
potential vanish at /> = a means that k can take on only those special 
values : 

fc WM = ^, n = 1,2,3,... (3.107) 

where jc mw are the roots ofJ m (x mn ) = 0. 

Combining all these conditions, we find that the general form of the 
solution is 

® (p, <M = 2 1 JmiKnP) sinh {k mn z)[A mn sin m<f> + B mn cos m<f>] 

m=0 n = l 

(3.108) 
At z = L, we are given the potential as V(p, <f>). Therefore we have 
V(p> <f>) = 2 sinn (.k mn L)J m (k mnP )[A mn sin m<f> + B mn cos w0] 

m,n 

This is a Fourier series in <f> and a Bessel-Fourier series in p. The coeffi- 
cients are, from (2.43) and (3.97), 



_ 2 cosech 

mn ,2 r2 



and 



TTCfJ 

2 cosech 



Bwm Tra 2 ^ 



;h(kwwL ^ f *<ty f °dp pF(p, WJkmnP) sin m<£ 

+ l( fc «n fl ) ■'O *>0 

:ll(fcmwL) P^ f'dp PHP, WmikmnP) COS m<f> 

+i(k mn a) Jo Jo 



(3.109) 



with the proviso that, for m = 0, we use fB^ in the series. 

The particular form of expansion (3.108) is indicated by the requirement 
that the potential vanish at z = for arbitrary p and at p = a for arbitrary 
z. For different boundary conditions the expansion would take a different 



[Sect. 3.8] Boundary-Value Problems in Electrostatics: II 11 

form. An example where the potential is zero on the end faces and equal 
to V(<f>, z) on the side surface is left as Problem 3.6 for the reader. 

The Fourier-Bessel series (3.108) is appropriate for a finite interval in 
P, < p < a. If a -> oo, the series goes over into an integral in a manner 
entirely analogous to the transition from a trigonometric Fourier series 
to a Fourier integral. Thus, for example, if the potential in charge-free 
space is finite for z > and vanishes for z — ► oo, the general form of the 
solution for z > must be 

oo /» 

<D0, <f>, z) = y \dk e- kz JJk P )[AJk) sin m<f> + BJk) cos m<£] (3.110) 

If the potential is specified over the whole plane z = to be V(p, <f>) the 
coefficients are determined by 

oo <» 

v (p, <f>) = y\ dkJJkp)[A m (k) sin m<f> + BJk) cos m<f\ 

The variation in (f> is just a Fourier series. Consequently the coefficients 
A m (k) and B m (k) are separately specified by the integral relations : 

WJo Icosm^J Jo U m (fe')J 

These radial integral equations of the first kind can be easily solved, since 
they are Hankel transforms. For our purposes, the integral relation, 

xJJkx)JJk'x) dx = \ d(k' - k) (3.112) 

Jo k 

can be exploited to invert equations (3.111). Multiplying both sides by 
P J J^p) and integrating over p, we find with the help of (3.112) that the 
coefficients are determined by integrals over the whole area of the plane 
z = 0: 

A m( k )} kC 00 C 2 * fsinm<A ^^^ 

R J = £ d P p \ WfaWJlW \ (3 ' 113) 

BmW) nJ o Jo [cos m(f> 

As usual, for m = 0, we must use $B (k) in series (3.110). 



3.8 Expansion of Green's Functions in Spherical Coordinates 

In order to handle problems involving distributions of charge as well as 
boundary values for the potential (i.e., solutions of Poisson's equation) it 
is necessary to determine the Green's function G(x, x') which satisfies the 



78 



Classical Electrodynamics 



appropriate boundary conditions. Often these boundary conditions are 
specified on surfaces of some separable coordinate system, e.g., spherical or 
cylindrical boundaries. Then it is convenient to express the Green's 
function as a series of products of the functions appropriate to the coordi- 
nates in question. We first illustrate the type of expansion involved by 
considering spherical coordinates. 

For the case of no boundary surfaces, except at infinity, we already 
have the expansion of the Green's function, namely (3.70) : 



x — x 



00 _ I 

=4 "2 2 



1 



1 = m = -I 



21 + 1 r\ 



Y?JP'> Wm(0» 



Suppose that we wish to obtain a similar expansion for the Green's 
function appropriate for the "exterior" problem with a spherical boundary 
at r = a. The result is readily found from the image form of the Green's 
function (2.22). Using expansion (3.70) for both terms in (2.22), we obtain : 



G(x, x') = 4tt y — !— 
K ' f" 2/ + 1 

l,m ' 



-r 1 ? 1 a\rr J J 



To see clearly the structure of (3.114) and to verify that it satisfies the 
boundary conditions, we exhibit the radial factors separately for r < r' 
and for r > /: 

„2l + l~ 



1 



_rl +1 a\rr'J 



2\* + l- 



r — 



j+i 



r <r' 



(3.115) 



3Ti» r>r ' 



First of all, we note that for either r or r' equal to a the radial factor 
vanishes, as required. Similarly, as rorr'^ oo, the radial factor vanishes. 
It is symmetric in r and r' . Viewed as a function of r, for fixed r\ the 
radial factor is just a linear combination of the solutions r l and r~ {l+1) of the 
radial part (3.7) of Laplace's equation. It is admittedly a different linear 
combination for r < r' and for r > r' . The reason for this will become 
apparent below, and is connected with the fact that the Green's function 
is a solution of Poisson's equation with a delta function inhomogeneity. 
Now that we have seen the general structure of the expansion of a 
Green's function in separable coordinates we turn to the systematic con- 
struction of such expansions from first principles. A Green's function for 
a potential problem satisfies the equation 



W x 2 G(x, x') = -4tt <5(x - x') 



(3.116) 



[Sect. 3.8] Boundary-Value Problems in Electrostatics: H 79 

subject to the boundary conditions G(x, x') = for either x or x' on the 
boundary surface S. For spherical boundary surfaces we desire an expan- 
sion of the general form (3.1 14). Accordingly we exploit the fact that the 
delta function can be written * 

d(x - x') = \ d(r - r') d(<f> - <f>') <3(cos 6 - cos 0') 
r 2 

and that the completeness relation (3.56) can be used to represent the 
angular delta functions: 

.. oo I 

d(x-x') = \d(r-r')^ J YtnW'.WtJQ,*) ( 3 - 117 ) 

1 = m= -I 

Then the Green's function, considered as a function of x, can be expanded 
as 

oo I 

G(x, x') = I I A lm (6', <f>') gl (r, r')Y lm (e, <f>) (3.H8) 

1 = m= —l 

Substitution of (3.117) and (3.118) into (3.116) leads to the results 

A lm {e',<f>')=Y l l{d'^') (3.119) 

and 

Id* ( /Yk /(/ + 1) , ,. 4tt- 

- — 2 (rgi(r, r')) - ^— J gl (r, r ) = - - 
r dr 2 r* r 



" 4" 2 (rgfr, r')) - ^-±^ gl (r, V) = - ^ d(r - r') (3.120) 



The radial Green's function is seen to satisfy the homogeneous radial 
equation (3.7) for r ^ r ' . Thus it can be written as : 

(Ar l + Br~ il+1 \ for r < r' 

gi(r, r ) = ^ + B > r -«+i\ for r > r' 

The coefficients A, B, A', B' are functions of r' to be determined by the 
boundary conditions, the requirement implied by d(r — r') in (3.120), and 
the symmetry of gl (r, r') in r and r'. Suppose that the boundary sufaces are 
concentric spheres at r = a and r = b. The vanishing of G(x, x') for x on 

* To express d(x - x') = <5(*i - *i') %2 - *■') K x z — x s) in terms of the coordi- 
nates (li, l 2 , l 3 ), related to (x u x 2 , x 3 ) via the Jacobian J(x u £*), we note that the mean- 
ingful quantity is <5(x — x') d 3 x. Hence 

d(x - x') = n-^-r-7 <K£i - Si') <Kf . - f .0 <5(S 3 - f .0 



80 



Classical Electrodynamics 



the surface implies the vanishing of g x {r, r') for r = a and r = b. Con- 
sequently g t (r, r') becomes 



gi(r, r') = 



a 2l+1 \ 
r i + i)> 

B ,(J rM 

\ r l + l b 2l + lJ> 



r <r' 

r>r' 



(3.121) 



The symmetry in r and r' requires that the coefficients A(r') and B'(r') be 
such that g t (r, r') can be written 

where r< (>•.>) is the smaller (larger) of r and r'. To determine the constant 
C we must consider the effect of the delta function in (3. 1 20). If we multiply 
both sides by r and integrate over the interval from r = r' — e to r = r' + 
e, where e is very small, we obtain 



■f(r gl (r,r')) 
Ldr 



^{rgi(r,r>)) 

r' + e Ldr 



Ait 



= - - (3.123) 



Thus there is a discontinuity in slope at r = r', as indicated in Fig. 3.8. 
For r = r' + e, r > = r, r < = r' . Hence 



J {rgir, r')) 
Ldr 



Similarly 



T-(^(r,r')) 
Ldr 



_r{, a n+x \\dl\ r' +1 \" 
\ r' l+l /ldr\r l b 2l+1 I l=r> 

-^-fen('«-<n 
^♦■♦■fen-en 



Z + 1 + /I-I 111 

r'-e r ^ 

Substituting these derivatives into (3.123), we find: 

4tt 



C = 



(2/ + 1) 



Hfl 



(3.124) 



Combination of (3.124), (3.122), (3.119), and (3.118) yields the expansion 
of the Green's function for a spherical shell bounded by r = a and r = b: 



z=o m=-f (2/ + 1) 



Mi)' 



[Sect. 3.9] Boundary-Value Problems in Electrostatics: II 



81 



Fig. 3.8 Discontinuity in slope of 
the radial Green's function. 





\ 1 




\ ' 




x 1 ' 


k 


\ 




\ 1 y 




Jlv^ 


V. 


-<^l\v 


C 








1 


^ ' \^*^^ 




1 X 



For the special cases a -+ 0, b -> oo, and b -> oo, we recover the previous 
expansions (3.70) and (3.114), respectively. For the "interior" problem 
with a sphere of radius b we merely let a -► 0. Whereas the expansion for 
a single sphere is most easily obtained from the image solution, the general 
result (3.125) for a spherical shell is rather difficult to obtain by the method 
of images, since it involves an infinite set of images. 



3.9 Solution of Potential Problems with the Spherical Green's Function 
Expansion 

The general solution to Poisson's equation with specified values of the 
potential on the boundary surface is (see Section 1.10): 

O(x) = I p(x')G(x, x') dV - - ct O(x') ^ da' (3.126) 
J v Att Js on' 

For purposes of illustration let us consider the potential inside a sphere of 
radius b. First we will establish the equivalence of the surface integral in 
(3.126) to the previous method of Section 3.4, equations (3.61) and (3.58). 
With a = in (3.125), the normal derivative, evaluated at r' = b, is: 



dG 

dn' 



dG 

dr' 



,_ 6 = ~ j^{i) 1yU6 '' <m ™ (0 ' ^ (3 - 127) 



l,m 



Consequently the solution of Laplace's equation inside r = b with 
$ = V{0', ^') on the surface is, according to (3.126): 



<D(x) 



l,m 



)YtJd',<f>')dO.' 



(;)' 



Y lm (d, <f>) (3.128) 



For the case considered, this is the same form of solution as (3.61) with 
(3.58). There is a third form of solution for the sphere, the so-called 



82 



Classical Electrodynamics 




Fig. 3.9 Ring of charge of radius a and 
total charge Q inside a grounded, conduct- 
ing sphere of radius b. 

Poisson integral (2.25). The equivalence of this solution to the Green's 
function expansion solution is implied by the fact that both were derived 
from the general expression (3.126) and the image Green's function. The 
explicit demonstration of the equivalence of (2.25) and the series solution 
(3.61) will be left to the problems. 

We now turn to the solution of problems with charge distributed in the 
volume, so that the volume integral in (3.126) is involved. It is sufficient 
to consider problems in which the potential vanishes on the boundary 
surfaces. By linear superposition of a solution of Laplace's equation the 
general situation can be obtained. The first illustration is that of a hollow 
grounded sphere of radius b with a concentric ring of charge of radius a 
and total charge Q. The ring of charge is located in the x-y plane, as shown 
in Fig. 3.9. The charge density of the ring can be written with the help of 
delta functions in angle and radius as 

p(x') = -£- d(r' - a) «5(cos 6>') (3.129) 

27ur 

In the volume integral over the Green's function only terms in (3.125) with 
m = will survive because of azimuthal symmetry. Then, using (3.57) 
and remembering that a -+ in (3.125), we find 

O(x) = J />(x')G(x, x') d 3 x' 

i=o Xr> ° ' 

where now r < (r>) is the smaller (larger) of r and a. Using the fact that 

(— l) n (2n — 1)" 
An+iCO) = and P 2n (0) = ^ — — , (3.130) can be written as: 



<D(x) = q2 



2 n n\ 
■l)"(2n-l)l! 2w 



2 n n\ 



{-^-■0r) P ^osB) (3.131) 



[Sect. 3.9] Boundary-Value Problems in Electrostatics: II 



83 




Fig. 3.10 Uniform line charge of 

length lb and total charge Q inside 

a grounded, conducting sphere of 

radius b. 

In the limit b^ oo, it will be seen that (3.130) or (3.131) reduces to 
expression (3.48) for a ring of charge in free space. The present result can 
be obtained alternatively by using (3.48) and the images for a sphere. 

A second example of charge densities, illustrated in Fig. 3.10, is that of 
a hollow grounded sphere with a uniform line charge of total charge Q 
located on the z axis between the north and south poles of the sphere. 
Again with the help of delta functions the volume-charge density can be 
written : 

p(x') = Q _1_. [5( C os 6' - 1) + <5(cos 0' + 1)] (3.132) 

2b 2-nr'"' 

The two delta functions in cos correspond to the two halves of the line 
charge, above and below the x-y plane. The factor 2-nr" 1 in the denominator 
assures that the charge density has a constant linear density Q\2b. With 
this density in (3.126) we obtain 

*(*) = Tu 2 CW) + p i(~ W p i( cos e "> [ r <{-7^ ~ iiri) dr ' (3 - 133) 

The integral must be broken up into the intervals < r' < r and 
r </ < b. Then we find 



1(1 +> 1)\ \b 
For / = this result is indeterminate 
have, for / = only, 

d 

dl\ \b 



(3.134) 
Applying L'Hospital's rule, we 



P_ to ^LJeI _ Um (_ I />»<*) = ,„ (*) (3.135) 
Jo i-o d ... i-o \ dl J \rl 



dl 



84 



Classical Electrodynamics 



This can be verified by direct integration in (3.133) for / = 0. Using the 
fact that Pj(-l) = (-1)', the potential (3.133) can be put in the form: 



<D(x) 



?M9 



+ 



V (4/ + 1) 
^2;(2/- + l) 



1 - (y P*fro& 0)} (3.136) 



The presence of the logarithm for / = reminds us that the potential 
diverges along the z axis. This is borne out by the series in (3.136), which 
diverges for cos 6 = ±1, except at r = b exactly. 

The surface-charge density on the grounded sphere is readily obtained 
from (3.136) by differentiation: 



«(9) -i-?* 

477 dr 



Antfl 



1 + y(4L±i)p 2Xcos0) 
£i(2/ + D 



(3.137) 

The leading term shows that the total charge induced on the sphere is — Q, 
the other terms integrating to zero over the surface of the sphere. 



3.10 Expansion of Green's Functions in Cylindrical Coordinates 

The expansion of the potential of a unit point charge in cylindrical 
coordinates affords another useful example of Green's function expan- 
sions. We will present the initial steps in general enough fashion that the 
procedure can be readily adapted to finding Green's functions for potential 
problems with cylindrical boundary surfaces. The starting point is the 
equation for the Green's function : 

V, 2 G(x, X ') = - — d(p - />') d(<j> - 0') d(z - z') (3.138) 

P 

where the delta function has been expressed in cylindrical coordinates. 
The <f> and z delta functions can be written in terms of orthonormal 
functions : 

d(z - z') = — dk e ik{z - z,) = - \ dk cos [k(z - 2')] 

2.7T J — 00 77 Jo 

1 °° . 

d(<f> —(/>') = — V £<«(♦-♦') 



m= — 00 



(3.139) 



We expand the Green's function in similar fashion : 

G(x,x') = — ^ )dke in **-^coslk{z-z')]g m {p,p') (3.140) 

«i = — on ^ 



85 



[Sect. 3.10] Boundary-Value Problems in Electrostatics: II 

Then substitution into (3.138) leads to an equation for the radial Green's 
function g m (p, p'): 

1 t i? ¥) - i k * + *)*- = - - «* - * < 3 - i4i) 

p dp\ dp l \ pi p 

For p # p this is just equation (3.98) for the modified Bessel functions, 
IJkp) and KJkp). Suppose that yjjep) is some linear combination of 
I m and K m which satisfies the correct boundary conditions for p < p, and 
that y) 2 (kp) is a linearly independent combination which satisfies the 
proper boundary conditions for p > p. Then the symmetry of the Green's 
function in /> and p requires that 



gm(p, P') = V>i(kp<)y>2(kp>) 



(3.142) 



The normalization of the product ^ 2 is determined by the discontinuity 
in slope implied by the delta function in (3.141): 



dp 



dp 



Att 

7 



(3.143) 



where I . means evaluated at p = p ± e. From (3.142) it is evident that 



= Kv>iV>* - VWi') = k^l>i> Vs] (3.144) 



dg m 

- dp 


_ dgrn 

+ dp 





where primes mean differentiation with respect to the argument, and 
W{(p x ,ip^ is the Wronskian of ^ and ip 2 . Equation (3.141) is of the 
Sturm-Liouville type 






(3.145) 



and it is well known that the Wronskian of two linearly independent 
solutions of such an equation is proportional to [l/p(x)]. Hence the 
possibility of satisfying (3.143) for all values of p is assured. Clearly we 
must demand that the normalization of the product %p x tp 2 is such that the 
Wronskian has the value : 



x 



(3.146) 



If there are no boundary surfaces, the requirement is that g m (p, p) be 
finite at /> = and vanish at p -> oo. Consequently tp x (kp) = AI m (kp) and 
ip 2 (fcp) = KJJcp). The constant A is to be determined from the Wronskian 
condition (3.146). Since the Wronskian is proportional to (\jx) for all 
values of x, it does not matter where we evaluate it. Using the limiting 



86 Classical Electrodynamics 

forms (3.102) and (3.103) for small x [or (3.104) for large x], we find 

W[I m (x), K m (z)] = - i (3.147) 

x 

so that A = Att. The expansion of l/|x — x'| therefore becomes: 
-J— - = l J \*dk *«**-*"> cos [k(z-z')-\IJk P< )K m (k P> ) (3.148) 
This can also be written entirely in terms of real functions as : 

i 4 r°° 

= -\ dk cos [k(z - z')~] 

|x — x'| IT Jo 

x hl (k P< )K (k P> ) + y cos [m(<£ - f )]J M (fe/»<)X w (kp>) 



(3.149) 

A number of useful mathematical results can be obtained from this 
expansion. If we let x' — >■ 0, only the m = term survives, and we obtain 
the integral representation : 

i 2 r°° 

-j=L= = - cos kz K (k P ) dk (3.150) 

V f> 2 + Z 2 TT J <> 

If we replace /> 2 in (3.150) by R 2 = P 2 + />' 2 - 2 PP cos (0 - <f>'\ then we 
have on the left-hand side the inverse distance |x — x'| _1 with z' = 0, i.e., 
just (3.149) with z = 0. Then comparison of the right-hand sides of 
(3.149) and (3.150) (which must hold for all values of z) leads to the 
identification : 



K (kJ P 2 + P ' 2 - 2 PP ' cos (<f> - #) ) 

= I (k P< )K (k P> ) + 2^008 [m(<J> - cf>')-]I m (k P< )K m (k P> ) (3.151) 

m = l 

In this last result we can take the limit k —> and obtain an expansion for 
the Green's function for (two-dimensional) polar coordinates : 

1 



In 



^p 2 + p' 2 - 2pp' cos (tp - <p')j 

= In (— ) + y -(^fcos [m(<£ - f )] (3.152) 

This representation can be verified by a systematic construction of the 
two-dimensional Green's function for Poisson's equation along the lines 
leading to (3.148). 



[Sect. 3.11] Boundary- Value Problems in Electrostatics: II 87 

3.11 Eigenfunction Expansions for Green's Functions 

Another technique for obtaining expansions of Green's functions is the 
use of eigenf unctions for some related problem. This approach is inti- 
mately connected with the methods of Sections 3.8 and 3.10. 

To specify what we mean by eigenfunctions, we consider an elliptic 
differential equation of the form : 

VVx) + [/(x) + Afy(x) = (3.153) 

If the solutions y(x) are required to satisfy certain boundary conditions 
on the surface S of the volume of interest V, then (3.153) will not in general 
have well-behaved (e.g., finite and continuous) solutions, except for 
certain values of A. These values of A, denoted by A n , are called eigenvalues 
(or characteristic values) and the solutions tpjx) are called eigenfunctions.* 
The eigenvalue differential equation is written: 

VV(x) + [/(x) + A> n (x) = (3.154) 

By methods similar to those used to prove the orthogonality of the 
Legendre or Bessel functions it can be shown that the eigenfunctions are 
orthogonal : 

I* </V*(x)Vn(x) <?x = d mn (3.155) 

Jv 

where the eigenfunctions are assumed normalized. The spectrum of 
eigenvalues A w may be a discrete set, or a continuum, or both. It will be 
assumed that the totality of eigenfunctions forms a complete set. 

Suppose now that we wish to find the Green's function for the equation : 

V x 2 G(x, x') + [fix) + A]G(x, x') = -4tt<5(x - x') (3.156) 

where A is not in general one of the eigenvalues A n of (3. 1 54). Furthermore, 
suppose that the Green's function is to have the same boundary conditions 
as the eigenfunctions of (3.154). Then the Green's function can be 
expanded in a series of the eigenfunctions of the form : 

G(x,x') = 2«n(*>»(x) (3.157) 

n 

Substitution into the differential equation for the Green's function leads 
to the result: 

2 a m (x')(A - A „> m (x) = -4tt<5(x - x') (3.158) 

m 

* The reader familiar with wave mechanics will recognize (3.153) as equivalent to the 
Schrodinger equation for a particle in a potential. 



88 Classical Electrodynamics 

If we multiply both sides by y> n *(x) and integrate over the volume V, the 
orthogonality condition (3.155) reduces the left-hand side to one term, and 
we find : * 

a n (x') = ^f^ (3.159) 

Consequently the eigenfunction expansion of the Green's function is : 

Gfr»o-^g * , yy (3 - i6o> 

n n 

For a continuous spectrum the sum is replaced by an integral. 

Specializing the above considerations to Poisson's equation, we place 
/(x) = and X = in (3.156). As a first, essentially trivial, illustration 
we let (3.154) be the wave equation over all space: 

(V 2 + k 2 )y k (x) = (3.161) 

with the continuum of eigenvalues, k 2 , and the eigenfunctions : 

1 Jkx 

(2*)' 

These eigenfunctions have delta function normalization : 



Vk(x) = 7r3s« ( 3162 > 



/ 



Vk'*(x)Vk(x) d 3 x = «5(k - k') (3.163) 

Then, according to (3.160), the infinite space Green's function has the 
expansion: - ik . (x - x <) 

i — = _L \d 3 k e —— (3.164) 

|x-x'| 2tt 2 J fc 2 

This is just the three-dimensional Fourier integral representation of 
l/|x-x'|. 

As a second example, consider the Green's function for a Dirichlet 
problem inside a rectangular box defined by the six planes, x = y = z = 0, 
x = a, y = b, zj*£ c. The expansion is to be made in terms of eigen- 
functions of the wave equation : 

F 2 + kf mn )y lmn (x,y,z) = (3.165) 

where the eigenfunctions which vanish on all the boundary surfaces are 



and 



, a / 8 . 1\itx\ . im-nyX . / 

VWO, y,*) = *]—r sm y— J sin \-y-J sin y 

lmn la 2 b 2+ cV 



mrz\ 
c I 



(3.166) 



[Sect. 3.12] Boundary-Value Problems in Electrostatics: II 89 

The expansion of the Green's function is therefore: 

oo 



I ,m,n — 1 



sin 
x 



. (Ittx\ . (Ittx'\ . [mTTy\ . lrmTy'\ . (nirz\ . jniTz'\ 

m U sin [—) sm Vf) sm \—) sm VT) sm VT) 



(i 2 + - 2 + !L 2 ) 
\a 2 b 2 cV 



(3.167) 

To relate expansion (3.167) to the type of expansions obtained in 
Sections 3.8 and 3.10, namely, (3.125) for spherical coordinates and 
(3.148) for cylindrical coordinates, we write down the analogous expansion 
for the rectangular box. If the x and y coordinates are treated in the 
manner of (6, <f>) or {<f>, z) in those cases, while the z coordinate is singled 
out for special treatment, we obtain the Green's function : 

16tt V • [lirx\ • (Ittx'\ . lrmry\ . (miry' 



l,m = l 

~ sinh (K lm z<) sinh (K lm (c - z>)) 
K lm sinh (K lm c) 



(3.168) 



must 



/ / 2 m 2 \ A 
where K lm = A — + — I . If (3.167) and (3.168) are to be equal, it 

be that the sum over n in (3.167) is just the Fourier series representation 

on the interval (0, c) of the one-dimensional Green's function in z in 

(3.168): , A 

,„ sm I 1 

sinh (K lmZ< ) sinh (K lm (c - z>)) = 2V \ c 1 ^ Imrz 

K lm sinh (K lm c) c£[ R ^ + jWj 2 \ c 

(3.169) 

The verification that (3.169) is the correct Fourier representation is left as 
an exercise for the reader. 

Further illustrations of this technique will be found in the problems at 
the end of the chapter. 



3.12 Mixed Boundary Conditions; Charged Conducting Disc 

The potential problems discussed so far in this chapter have been of the 
orthodox kind in which the boundary conditions are of one type (usually 
Dirichlet) over the whole boundary surface. In the uniqueness proof for 



90 Classical Electrodynamics 



^■■fcj 




Fig. 3.11 

solutions of Laplace's or Poisson's equation (Section 1.9) it was pointed 
out, however, that mixed boundary conditions, where the potential is 
specified over part of the boundary and its normal derivative is specified 
over the remainder, also lead to well-defined, unique, boundary-value 
problems. There is a tendency in existing textbooks to mention the 
possibility of mixed boundary conditions when making the uniqueness 
proof and to ignore such problems in subsequent discussion. The reason, 
as we shall see immediately, is that mixed boundary conditions are much 
more difficult to handle than the normal type. 

To illustrate the difficulties encountered with mixed boundary con- 
ditions we consider the apparently simple problem of an isolated, infinitely 
thin, flat, circular, conducting disc of radius a with a total charge q placed 
on it, as shown in Fig. 3.1 1. The charge distributes itself over the disc in 
such a way as to make its surface an equipotential. We wish to determine 
the potential everywhere in space and the charge distribution on the disc. 

From the geometry of the problem we see that the potential is symmetric 
about the axis of the disc and with respect to the plane containing the disc. 
If cylindrical coordinates are chosen with the axis of the disc as the z axis 
and the origin at the center of the disc, the potential must therefore be of 
the form [from (3.110)], 

(D( P , z) = ( X dkf(k)e- k]zl J (k P ) (3.170) 

Jo 

The unknown function f{k) must be determined from the boundary 
conditions at z = 0. If the potential were known everywhere over the 
whole z = plane,/(&) could be found by inverting the Hankel transform, 
as in going from (3.110) to (3.113). Unfortunately the boundary con- 
ditions at z = are not that simple. For < p < a we do know that the 
potential is constant at an unknown value, O = V = q/C, where C is the 
capacitance of the disc. But for a < p < oo, the potential is unknown. 



{Sect. 3.12] Boundary-Value Problems in Electrostatics: II 91 

From symmetry, however, we know that the normal derivative of the 
potential vanishes there. Thus the boundary conditions are mixed: 

0>O, 0) = V, for < P < a 



^0>,0) = 0, 

oz 



for a < p < oo 



(3.171) 



The connection between the potential of the disc V and the total charge q 
on it will be established by the fact that at large distances (p and/or z > a) 
the potential must approach qj(p 2 + z 2 ) v K From (3.170) and an identity 
of Problem 3.12c this requirement can be seen to imply 

\imf(k)=q (3.172) 

fc->0 

When boundary conditions (3.171) are applied to the general solution 
(3.170), there results a pair of integral equations of the first kind: 






dkf(k)J (kp) =V, for < P 



dk kf(k)J (kp) = 0, for a < p < oo 



(3.173) 



Such pairs of integral equations, with one of the pair holding over one 
part of the range of the independent variable and the other over the other 
part of the range, are known as dual integral equations. The general theory 
of dual integral equations is complicated and not highly developed. But 
the charged disc problem and variations of it have received considerable 
attention over the years. H. Weber (1873) first solved the present problem 
by using certain discontinuous integrals involving Bessel functions. 
Titchmarsh, p. 334, uses Mellin transforms to effect a solution of a some- 
what more general pair of dual integral equations. E. T. Copson [Proc. 
Edin. Math. Soc. (2), 8, 14 (1947)] reduces the disc problem to an integral 
equation for the surface-charge density of the Abel type. Tranter, p. 50 
and Chapter VIII, considers slight generalizations of the pair (3.173). He 
introduces a systematic technique of finding the most general form satis- 
fying the homogeneous member of the pair and then delimiting that form 
by substitution into the other equation. The Wiener-Hopf technique can 
also be used. 

For our purposes it is sufficient to observe that the dual integral 
equations, 



r 



dyg{y)J n {yx) = x r > 



dy y g(y)J n (y x ) = 0> 



for < x < l 



for 1 < x < oo 



(3.174) 



92 

have the solution, 



Classical Electrodynamics 



*)--£^^>-j^(fw») (3-175) 

In this relation y' w (?/) is the spherical Bessel function of order n (see Section 
16.1). For the set of equations (3.173) the variables are x = p/a and 
y = ka, while n = 0. Thus the solution is 

f(k) = -Va j (ka) = - Va( S ^) (3.176) 

77 7T \ ka / 

Remembering the connection (3.172) which determines the potential K 
in terms of the charge q, we find 

2 a 
This shows that the capacitance of a disc of radius a is 

r 2 
C = - a 



This value was experimentally established with remarkable precision by 
Cavendish (ca. 1780) by comparing the charges on a disc and a sphere at 
the same potential. 
The potential anywhere in space is found from (3.170) and (3.176) to be 



*0> 



Jo 



00 ,, sin ka _ fr 
dk - * 



/ca 



J (fc/>) 



(3.177) 



Values of the potential along the axis and in the plane of the disc can be 
found readily by putting p = and z = in (3.177). The results are 



O(0, z) = - tan 
a 



-l 



(!) 



$(/>, 0) = - 



- sin x 
a 

77 # 

2a' 



(!)■ 



for p > a 
for < p < a 



For arbitrary p and z the integral can be transformed into Weber's form 
of the solution : 

2a 



<X>(p, z) = q sin" 



-V(/> - a) 2 + z 2 + V(p + a) 2 + z 2 -l 



(3.178) 



[Sect. 3.12] Boundary-Value Problems in Electrostatics: II 93 

The charge density a(p) on the surface of the disc is given by 

a(p) = - — — 0, 0) = -^- dk sin ka J (kp) 
2tt dz lira Jo 

The integral is a well-known discontinuous integral which vanishes for 
p > a. For p < a, the charge density is 

o(p) = -2 , 1 (3.179) 

lira Vfl 2 - p 2 

The (integrable) infinity in a(p) for p -► a is a mathematical singularity 
which results from the assumption of an infinitely thin disc. In practice 
the charge is repelled to the outer regions of a thin disc approximately 
according to (3.179), but near the edge the distribution levels off to a large, 
but finite, value which depends on the detailed construction of the disc. 
We have discussed the charged conducting disc in cylindrical coordinates 
in order to illustrate the complications of mixed boundary conditions. 
For this particular example, the mixed boundary conditions can be avoided 
by separating Laplace's equation in elliptic coordinates. Then the disc 
can be taken to be the limiting form of an oblate spheroidal surface. See, 
for example, Smythe, pp. Ill, 156, or Jeans, p. 244. 

REFERENCES AND SUGGESTED READING 

The mathematical apparatus and special functions needed for the solution of potential 
problems in spherical, cylindrical, spheroidal, and other coordinate systems are discussed 
in 

Morse and Feshbach, Chapter 10. 
A more elementary treatment, with well-chosen examples and problems, can be found in 

Hildebrand, Chapters 4, 5, and 8. 
A somewhat old-fashioned source of the theory and practice of Legendre polynomials 
and spherical harmonics, with many examples and problems, is 

Byerly. 
For purely mathematical properties of spherical functions one of the most useful 
one-volume references is 

Magnus and Oberhettinger. 
For more detailed mathematical properties, see 

Watson, for Bessel functions, 

Bateman Manuscript Project books, for all types of special functions. 
Electrostatic problems in cylindrical, spherical, and other coordinates are discussed 
extensively in 

Durand, Chapter XI, 

Jeans, Chapter VIII, 

Smythe, Chapter V, 

Stratton, Chapter III. 



94 Classical Electrodynamics 



PROBLEMS 

3.1 The surface of a hollow conducting sphere of inner radius a is divided into 
an even number of equal segments by a set of planes whose common line of 
intersection is the z axis and which are distributed uniformly in the angle </>. 
(The segments are like the skin on wedges of an apple, or the earth's 
surface between successive meridians of longitude.) The segments are kept 
at fixed potentials ± V, alternately. 

(a) Set up a series representation for the potential inside the sphere for 
the general case of 2n segments, and carry the calculation of the coefficients 
in the series far enough to determine exactly which coefficients are different 
from zero. For the nonvanishing terms, exhibit the coefficients as an 
integral over cos 0. 

(b) For the special case of n = 1 (two hemispheres) determine explicitly 
the potential up to and including all terms with 1 = 3. By a coordinate 
transformation verify that this reduces to result (3.37) of Section 3.3. 

3.2 Two concentric spheres have radii a, b (b > a) and are divided into two 
hemispheres by the same horizontal plane. The upper hemisphere of the 
inner sphere and the lower hemisphere of the outer sphere are maintained 
at potential V. The other hemispheres are at zero potential. 

Determine the potential in the region a < r < b as a series in Legendre 
polynomials. Include terms at least up to / = 4. Check your solution 
against known results in the limiting cases b -> oo, and a -> 0. 

3.3 A spherical surface of radius R has charge uniformly distributed over its 
surface with a density QJAttR 2 , except for a spherical cap at the north pole, 
defined by the cone 6 = a. 

(a) Show that the potential inside the spherical surface can be expressed 
as 



ev i 



o =^ > 



2 Zw2/ + 

1=0 



- [P J+1 (cos a) - iV^COS a)] — ^(cos 0) 



where, for / = 0, P^cos a) = —1. What is the potential outside? 

(b) Find the magnitude and the direction of the electric field at the origin. 

(c) Discuss the limiting forms of the potential (a) and electric field (b) as 
the spherical cap becomes (1) very small, and (2) so large that the area 
with charge on it becomes a very small cap at the south pole. 

3.4 A thin, flat, conducting, circular disc of radius R is located in the x-y plane 
with its center at the origin, and is maintained at a fixed potential V. With 
the information that the charge density on a disc at fixed potential is 
proportional to (R 2 — p 2 )~ iA , where p is the distance out from the center of 
the disc, 

(a) show that for r > R the potential is 

1=0 

(b) find the potential for r < R. 



[Probs. 3] Boundary-Value Problems in Electrostatics: II 95 

3.5 A hollow sphere of inner radius a has the potential specified on its surface 
to be $> = V(d, <j>). Prove the equivalence of the two forms of solution for 
the potential inside the sphere: 

, ^ a(a 2 - r 2 ) f V(0', f ) dn , 

(a) <D(x) = ^ J (r i +fl ._2arcosy)» 

where cos y = cos cos 0' + sin sin 0' cos (<£ — <£')• 

(6) $(x) = 2 2 ^ m w y ' w(0 ' ^ 



i=0 OT=-I 



where ^ ?m = (dCl' Y* m (d', <t>')V(6' , <j>'). 



3.6 A hollow right circular cylinder of radius b has its axis coincident with the 
z axis and its ends at z = and z = L. The potential on the end faces is 
zero, while the potential on the cylindrical surface is given as V(<f>, z). 
Using the appropriate separation of variables in cylindrical coordinates, 
find a series solution for the potential anywhere inside the cylinder. 

3.7 For the cylinder in Problem 3.6 the cylindrical surface is made of two 
equal half-cylinders, one at potential V and the other at potential - V, so 
that 



V(<f>, z) = « 



IT , IT 

Kfor -- <<£ <- 

TT , 37T 

-Kfor- <<£ <y 



(a) Find the potential inside the cylinder. 

(b) Assuming L > b, consider the potential at z = L/2 as a function of P 
and <f> and compare it with two-dimensional Problem 2.8. 

3.8 Show that an arbitrary function fix) can be expanded on the interval 
< x < a in a modified Fourier-Bessel series 

fix) = 2_. A nJv\yvnA 
w = l 

where y m is the nth root ^p = 0, and the coefficients A n are given by 

K = ~t — ^T7~ S? (x)xj '{ y "' 3 * 

3.9 An infinite, thin, plane sheet of conducting material has a circular hole of 
radius a cut in it. A thin, flat disc of the same material and slightly smaller 
radius lies in the plane, filling the hole, but separated from the sheet by a 
very narrow insulating ring. The disc is maintained at a fixed potential V, 
while the infinite sheet is kept at zero potential. 

(a) Using appropriate cylindrical coordinates, find an integral expression 
involving Bessel functions for the potential at any point above the plane. 



96 Classical Electrodynamics 

(b) Show that the potential a perpendicular distance z above the center 
of the disc is 

® n (z) = 



o(z) = v(l - * \ 
\ Va 2 + z 2 / 



(c) Show that the potential a perpendicular distance z above the edge of 
the disc is 



*«(z) = \ 



1 ~^K{k) 
tt a 



where k = 2a/(z 2 + 4a 2 )^, and K(k) is the complete elliptic integral of the 
first kind. 

3.10 Solve for the potential in Problem 3.2, using the appropriate Green's 
function obtained in the text, and verify that the answer obtained in this 
way agrees with the direct solution from the differential equation. 

3.11 A line charge of length 2d with a total charge Q has a linear charge density 
varying as (d 2 — z 2 ), where z is the distance from the midpoint. A grounded, 
conducting, spherical shell of inner radius b > d is centered at the midpoint 
of the line charge. 

(a) Find the potential everywhere inside the spherical shell as an 
expansion in Legendre polynomials. 

(b) Calculate the surface-charge density induced on the shell. 

(c) Discuss your answers to (a) and {b) in the limit that d<^b. 

3.12 (a) Verify that 

1 f 00 

- <X/> -/>')= kJ m {kp)JJkp) dk 

P Jo 

(b) Obtain the following expansion : 

1 ^ f 00 

£TT^j- Z dke im ^-'l'V m (kp)J m (kp')e-^>-^ 

(c) By appropriate limiting procedures prove the following expansions : 

oo 

J (kV p +p' 2 -2 PP 'cos<f>) = ^ e im +JJJcp)J m (kp') 

m= — <x> 

CO 

e ik P cos 4 = V in e in*JJJc p ) 

m= -co 

{d) From the last result obtain an integral representation of the Bessel 
function : 

l C 2n 

J m (x) = \ e ix cos i-im^ty 

2iri m J 

Compare the standard integral representations. 



[Probs. 3] Boundary-Value Problems in Electrostatics: II 97 

3.13 A unit point charge is located at the point (?', 4>\ z') inside a grounded 
cylindrical box denned by the surfaces z = 0, z = L, P = a. Show that the 
potential inside the box can be expressed in the following alternative forms: 



oo oo ^*-*V m (^V«(— ) 

, ,. 4 V V \ a J \ a I 

x sinhp22z<lsinh 



122(1, -z>) 
a 



m=-°° m=l l m \——\ 



® (x ' x ' )= z^ 



>< 2 22- 

m= — a> fc = l »=1 



«(* -*'. sin fc) sin ^)/„(f^)/„^) 



M + 0' 



An +l\ x mn) 



Discuss the relation of the last expansion (with its extra summation) to the 
other two. 
3.14 The walls of the conducting cylindrical box of Problem 3.13 are all at zero 
potential, except for a disc in the upper end, denned by /> = b, at potential V. 

(a) Using the various forms of the Green's function obtained in Problem 
3.13, find three expansions for the potential inside the cylinder. 

(b) For each series, calculate numerically the ratio of the potential at 
P = 0, z = L/2 to the potential of the disc, assuming b = L/4 = a/2. Try 
to obtain at least two-significant-figure accuracy. Is one series less rapidly 
convergent than the others? Why? 

(Jahnke and Emde have tables of J and J x on pp. 156-163, I and I x on 
pp. 226-229, (2ln)K and (2fr)K x on pp. 236-243. Watson also has 
numerous tables.) 



4 



Multipoles, Electrostatics of 
Macroscopic Media, 
Dielectrics 



This chapter is first concerned with the potential due to localized 
charge distributions and its expansion in multipoles. The development is 
made in terms of spherical harmonics, but contact is established with the 
rectangular components for the first few multipoles. The energy of a 
multipole in an external field is then discussed. The macroscopic equations 
of electrostatics are derived by taking into account the response of atoms 
to an applied field and by suitable averaging procedures. Dielectrics and 
the appropriate boundary conditions are then described, and some 
typical boundary-value problems with dielectrics are solved. Simple 
classical models are used to illustrate the main features of atomic polariza- 
bility and susceptibility. Finally the question of electrostatic energy in the 
presence of dielectrics is discussed. 

4.1 Multipole Expansion 

A localized distribution of charge is described by the charge density 
p(x'), which is nonvanishing only inside a sphere of radius R* around some 
origin. The potential outside the sphere can be written as an expansion in 
spherical harmonics : 

, , V V 4tt Y lm (6, <f>) 

i=o m =-i ^ -r *- r 

* The sphere of radius R is an arbitrary conceptual device employed merely to divide 
space into regions with and without charge. 

98 



[Sect. 4.1] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 99 

where the particular choice of constant coefficients is made for later 
convenience. Equation (4.1) is called a multipole expansion; the / = 
term is called the monopole term, / = 1 is the dipole term, etc. The 
reason for these names becomes clear below. The problem to be solved 
is the determination of the constants q lm in terms of the properties of the 
charge density p(x'). The solution is very easily obtained from the 
integral (1.17) for the potential: 

J |x — x | 

with expansion (3.70) for l/|x - x'| . Since we are interested at the 
moment in the potential outside the charge distribution, r < = r' and 



r. Then we find: 



d)(x) = 4^2 iy-^j* 7 ** ( 6 '> <£»( x ') d * x '_ 

l,m 



Y lm (6, <f>) 
„i+i 



(4.2) 



(4.3) 



Consequently the coefficients in (4.1) are: 

qim=\Y l * m (d',cf>'y l p(x')d*x' 

These coefficients are called multipole moments. To see the physical inter- 
pretation of them we exhibit the first few explicitly in terms of cartesian 
coordinates : 



4oo 



= i JW)**'--^, 



4n 

Qio = 



= - fL \{x' - iy') P (x') d z x' = - /i- (p x - i Py ) 

N 877 J 'V 077 



jz'pV 



)d 3 x>=/±p 



Att 



(4.4) 



(4.5) 



(4.6) 



AN 2ttJ ll'S Ztt 

Only the moments with m > have been given, since (3.54) shows that for 
a real charge density the moments with m < are related through 



«i.-» = (-D m « 



im 



(4.7) 



100 



Classical Electrodynamics 



In equations (4.4)-(4.6), q is the total charge, or monopole moment, p is 
the electric dipole moment : 



p = j x>(x') dV 
and Q tj is the quadrupole moment tensor : 

<2„ = j(3x t 'z/ - r'%^p{x') d*x' 



(4.8) 



(4.9) 



We see that the /th multipole coefficients [(21 + 1) in number] are linear 
combinations of the corresponding multipoles expressed in rectangular 
coordinates. The expansion of <D(x) directly in rectangular coordinates : 

" p • X , 1 "'O _ X.Xi 



O(x) = ^ + 
r 



by direct Taylor's series expansion of l/|x — x'| will be left as an exercise 
for the reader. It becomes increasingly cumbersome to continue the 
expansion in (4.10) beyond the quadrupole terms. 

The electric field components for a given multipole can be expressed 
most easily in terms of spherical coordinates. The negative gradient of a 
term in (4.1) with definite /, m has spherical components: 



= Mi±j) Y lm (d, <f>) 

21 + 1 qim r l+2 



E e = - 



4-7T 



E^= - 



21+ 1 

4-7T 

21+ 1 



1 r) 

lim -j^i rr Y lm (d, (f>) 



dd 



111 



im 



sin 



Y lm(0, <f>) 



(4.11) 



d Y lm ldQ and Y lm {sm d can be expressed as linear combinations of other 

F Jm 's, but the expressions are not particularly illuminating and so will be 

omitted. The proper way to describe a vector multipole field is by vector 

spherical harmonics, discussed in Chapter 16. 

For a dipole p along the z axis, the fields in (4.11) reduce to the familiar 

form: 

2p cos 6 

„3 



£„ = 



E a = 



r 
p sin 6 



E^ = 



(4.12) 



These dipole fields can be written in vector form by recombining (4.12) or 
by directly operating with the gradient on the dipole term in (4.10). The 



[Sect. 4.2] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 101 
result for the field at a point x due to a dipole p at the point x' is: 

3n(p • n) - p 
|x - xf 
where n is a unit vector directed from x' to x. 



E(x) = J -ur"'-* (4.13) 

Ix - x' 3 



4.2 Multipole Expansion of the Energy of a Charge Distribution in an 
External Field 

If a localized charge distribution described by />(x) is placed in an 
external potential O(x), the electrostatic energy of the system is: 



W 



= fp(x)0(x) d 3 x (4.14) 



If the potential O is slowly varying over the region where p(x) is non- 
negligible, then it can be expanded in a Taylor's series around a suitably 
chosen origin: 

O(x) = O(0) + x • VO(0) + \ S Sx iXj ^~- (0) + • • • (4.15) 

2, ' t ' ( VX* OX j 

i ) 

Utilizing the definition of the electric field E = - VO, the last two terms 
can be rewritten. Then (4.15) becomes: 

O(x) = O(0) - x • E(0) - i 22*^' Ifr (0) + " ' 



% i 



Since V • E = for the external field, we can subtract 

ir 2 V • E(0) 
from the last term to obtain finally the expansion : 

o(x) = o(0) - x . e(0) - i 22 (3a: ^ " r2di3) iv (0) + " ' (4 ' 16) 



t 3 



When this is inserted into (4.14) and the definitions of total charge, dipole 
moment (4.8) and quadrupole moment (4.9), are employed, the energy 
takes the form: 

W = <?O(0) - P • E(0) - i y V(2 W ^ (0) + • • • (4.17) 

6 *—i *—* ox, 

i i l 

This expansion shows the characteristic way in which the various multi- 
poles interact with an external field — the charge with the potential, the 
dipole with the electric field, the quadrupole with the field gradient, and 
so on. 



102 Classical Electrodynamics 

In nuclear physics the quadrupole interaction is of particular interest. 
Atomic nuclei can possess electric quadrupole moments, and their magni- 
tudes and signs have a bearing on the forces between neutrons and protons, 
as well as the shapes of the nuclei themselves. The energy levels or states 
of a nucleus are described by the quantum numbers of total angular 
momentum / and its projection M along the z axis, as well as others which 
we will denote by a general index a. A given nuclear state has associated 
with it a quantum-mechanical charge density* p JM <k*)-> which depends 
on the quantum numbers (/, M, a), but which is cylindrically symmetric 
about the z axis. Thus the only nonvanishing quadrupole moment is q 20 
in (4.6), or Q 33 in (4.9).f The quadrupole moment of a nuclear state is 
defined as the value of (l/e) Q^ with the charge density p JMx ( x )> where e 
is the protonic charge : 



- 1 .!**- 



Qj Ma = " J (3^ - r') PjMa (x) d*x (4.18) 

The dimensions of Qjm<x are consequently (length) 2 . Unless the circum- 
stances are exceptional (e.g., nuclei in atoms with completely closed 
electronic shells), nuclei are subjected to internal fields which possess field 
gradients in the neighborhood of the nuclei. Consequently, according to 
(4. 17), the energy of the nuclei will have a contribution from the quadrupole 
interaction. The states of different M value for the same J will have 
different quadrupole moments Q JM(X , and so a degeneracy in M value 
which may have existed will be removed by the quadrupole coupling to the 
"external" (crystal lattice, or molecular) electric field. Detection of these 
small energy differences by radiofrequency techniques allows the deter- 
mination of the quadrupole moment of the nucleus. { 

The interaction energy between two dipoles p x and p 2 can be obtained 
directly from (4.17) by using the dipole field (4.13). Thus, the mutual 
potential energy is 

w = P 1 P2-3(n.p 1 )(n.p 2 ) 
l*i - x 2 | 3 

where n is a unit vector in the direction (x x — x 2 ). The dipole-dipole 
interaction is attractive or repulsive, depending on the orientation of the 
dipoles. For fixed orientation and separation of the dipoles, the value of 

* See Blatt and Weisskopf, pp. 23 ff., for an elementary discussion of the quantum 
aspects of the problem. 

t Actually Q xx and Q 22 are different from zero, but are not independent of Q 33 , being 
given by e„ = Q 22 = —%Q 33 . 

+"The quadrupole moment of a nucleus," denoted by Q, is defined as the value of 
Q JMa in the state M = J. See Blatt and Weisskopf, he. cit. 



[Sect. 4.3] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 103 

the interaction, averaged over the relative positions of the dipoles, is zero. 
If the moments are generally parallel, attraction (repulsion) occurs when 
the moments are oriented more or less parallel (perpendicular) to the line 
joining their centers. For antiparallel moments the reverse is true. The 
extreme values of the potential energy are equal in magnitude. 



4.3 Macroscopic Electrostatics; Effects of Aggregates of Atoms 

The equations 

V • e = Atto ) 

(4.20) 
V x e = 0j 

govern electrostatic phenomena of all types, provided the "microscopic" 
electric field € is derived from the total "microscopic" charge density p. 
For problems with a few idealized point charges in the vicinity of mathe- 
matically defined boundary surfaces, equations (4.20) are quite acceptable. 
But there are many physical situations in which a complete specification 
of the problem in terms of individual charges would be impossible. Any 
problem involving fields in the presence of matter is a case in point. A 
macroscopic amount of matter has of the order of 10 23±3 charges in it, all 
of them in motion to a greater or lesser extent because of thermal agitation 
or zero point vibration. 

Setting aside the question of whether electrostatics can be relevant to a 
situation in which the charges are in incessant motion, let us consider the 
task of handling macroscopic problems with large numbers of atoms or 
molecules. Clearly the solution for the electric field : 

<x) = \GLZ*l pW fitf (4.21) 

J |x — x | d 

is not very suitable, since (a) it involves a charge density p which must 
specify the exact positions of very many charges, and (b) it fluctuates 
wildly as the observation point moves by only very small distances (of the 
order of atomic dimensions). Fortunately, for macroscopic electrostatics 
we do not want as detailed information as is contained in (4.21). We are 
content with averages of electric field strengths over regions of the order 
of 10~ 6 cm 3 (i.e., 10- 2 cm linear dimension) or greater. Since atomic 
volumes are of the order of 10~ 24 cm 3 , there are of the order of 10 18 or more 
atoms in the volumes of macroscopic interest. This means that the micro- 
scopic fluctuations will be entirely averaged out. We will wish to deal with 
an average c(x) and p'(x). The averages will be over a macroscopically 



104 Classical Electrodynamics 

small volume AV, large enough, however, to contain very many atoms or 
molecules : _ 

<e(x)> = -L e(x + I) d^ 

AV J A 



' AV 



AVJav ■— J 



(4.22) 



The averaged quantities are denoted by angle brackets ( ) ; the variable 
\ ranges over the volume A V. 

The averaging procedure now allows us to answer the question of 
whether it is legitimate to talk in static terms when the charges in matter 
are in thermal motion. At any instant of time the very many charges in 
the volume A V will be in all possible states of motion. An average over 
them at that instant will yield the same result as an average at some later 
instant of time. Hence, as far as the averaged quantities are concerned, 
it is legitimate to talk of static fields and charges.* Furthermore, the 
averaging can be done as if the atomic charges were fixed in space at the 
positions they have at some arbitrary instant. Hence the situation can be 
regarded as electrostatic even at the microscopic level for purposes of 
calculation. 

In the treatment of macroscopic electrostatics it is useful to break up 
the averaged charge density (p(x)) into two parts, one of which is the 
averaged charge of the atomic or molecular ions, or excess free charge 
placed in or on the macroscopic body, and the other of which is the 
induced or polarization charge. In the absence of external fields, atoms 
or molecules may or may not have electric dipole moments, but if they do, 
the moments are randomly oriented. In the presence of a field, the atoms 
become polarized (or their permanent moments tend to align with the 
field) and possess on the average a dipole moment These dipole moments 
can contribute to the averaged charge density (/>'( x ))- Since the induced 
dipole moments tend to be proportional to the applied field, we will find 
that the macroscopic version of (4.20) will involve only one constant to 
characterize the average polarizability of the medium involved. 

To see how the induced dipole moments enter the problem we first 
consider the microscopic field due to one molecule with center of mass at 
the point x, in Fig. 4. 1 while the observation point is at x. The molecular 
charge density is p/(x'), where x' is measured from the center of mass of 
the molecule. It should be noted that p- in general depends on the position 
of Xj of the molecule, since the distortion of the charge cloud depends on 
the local field present. The microscopic electric field due to the y'th 

* This ignores the very small (at room temperature) induction and radiation fields due 
to the acceleration of the charges in their thermal motion. 



[Sect. 4.3] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 105 



Fig. 4.1 A molecule with center of 
mass at x s gives a contribution to 
the potential at the point P with 
position x. The internal coordinate 
x' is measured from the center of 
mass. 



Molecule 




molecule is 



e,(x) 



Jmol 



/>/(*') 



|x — x, — x 



dV 



(4.23) 



For observation points outside the molecule we can expand in multipoles 
around the center of mass of the molecule. According to (4.10), this leads 
to r / i 

e J— + v/ 1 



6,(X) = -V 



L|x - x,- 



(^i)-—] 



(4.24) 



where 



e. = \ P /(x') d*x' 
Pi = f x' Pj '(x')d*x 



(4.25) 



are the molecular charge and the dipole moment, respectively. The 
quadrupole term in (4.10) could have been retained, but as long as the 
macroscopic variations of field occur over distances large compared to 
molecular dimensions it contributes negligibly to the averaged field 
relative to the dipole term. Both e j and p 3 are functions of the position of 
the molecule. 
To obtain the microscopic field due to all the molecules we sum over;: 



€(X) 



= - v 2[— — + p>- * ^.(i — ^ — r)l ( 4 - 26 > 

f*L|x-x,| \|x-x,|/J 

We now want to average according to (4.22) in order to obtain a macro- 
scopic field. To facilitate this averaging procedure we replace the discrete 
sum over the molecules by an integral by introducing apparently con- 
tinuous charge and polarization densities : 



Pmol(x) = 2 e A X - X >) 
i 

Wmol(x) = 2 P,<5(X - X,) 



(4.27) 



106 Classical Electrodynamics 

Then (4.26) can be rewritten formally as : 



:(x) = -V (d 3 x"\^^l + „ mol (x") • V"( 1 —)~ 

J L|x-x"| \|x-x"|/J 



(4.28) 



To illustrate the averaging process we consider the first term in (4.28). 
The averaged value is, by (4.22): 



< €l (x)> = -V 



-^- f d 3 z(d 3 x" 

.AV Jav J 



pmoljx") 



|x + 5 



X"U 



(4.29) 



where we have used the fact that differentiation and averaging can be 
interchanged. If the variable of integration x" is replaced by x" = x' + %, 
then 

< Cl (x)> = -V -L d^ \d 3 x> pmol(x ' + 5) 1 ( 4.30) 

LAV Jav J |x - x'| J 

The equality of (4.29) and (4.30) shows the obvious equivalence of averag- 
ing by means of moving the observation point around the volume AV 
centered at x and averaging by moving the integration point over the 
molecules in a volume A V centered around x'. From definition (4.27) it 
is clear that the integral of p mol over the volume AF at x' just adds up all 
the molecular charges e, inside A V: 

±-J WpmolV + 5) = -i- J 

AVJav ^V^ 

If the macroscopic density of molecules at x' is N(x') molecules per unit 
volume and (e mol (x')> is the average charge per molecule within the volume 
AFatx', then 

~ I ^ 3 ^moi(x' + J-) = N(x')<e mol (x')> (4.31) 

LA V J AV 

Now (4.30) can be written 

(6 1 (x)) = -vf iV(x/)(gm0l(X,) >^ 

J |x - x'l 

Exactly similar considerations can be made for the second term in (4.28). 
With the same definitions of averages we have 



e. 



j- I d*£n mol (x' + ?) = iV(x')<p m oi(x')> 



(4.32) 



Then the averaged form of (4.28) is given by: 

<€(x)> = -V (N(x'){ ( * mol(x y + <p mo i(x')> • V'(— I )) dV (4.33) 

J \ |x — x I \|x — x'\/) 



[Sect. 4.3] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 107 

To obtain the macroscopic equivalent of (4.20) we take the divergence of 
both sides. Recalling that V 2 (l/|x - x'|) = -4tt6(x - x'), we find: 

V • <€(x)> = 4tt fiV(x'){<emoi(x')><5(x - x') + <Pmoi(x')> • V'<3(x - x')} dV 

From the properties of the delta function (Section 1.2) it follows that 

V • <€(x)> = 47rJV(x)<e mol (x)> - 4ttV • (JV(x)<p mol (x)» (4.34) 

This is of the form of the first equation of (4.20) with the charge density />' 
replaced by two terms, the first being the average charge per unit volume 
of the molecules and the second being the polarization charge per unit 
volume. The presence of the divergence in the polarization-charge density 
seems very natural when one thinks of how this part of the charge density is 
created. If we consider a small volume in the medium, part of the charge 
inside that volume may be due to the net charges on the molecules. But 
there is a contribution arising from the polarization of the charge cloud of 
the molecules in an external field, since, for example, molecules whose 
charge once lay totally inside the volume may now have part of their 
charge cloud outside the volume in question. If the polarization is uniform 
over the space containing our small volume, then as much charge will be 
brought in through the surface of the volume as will leave it, and there will 
be no net effect. But if the polarization is not uniform, there can be a net 
increase or decrease of charge within the volume, as indicated schemati- 
cally in Fig. 4.2. This is the physical origin of the polarization-charge 
density. 

In (4.34) the two divergences can be combined so that the equation 

V • [(€> + 477iV(p mol >] = 47rN(e mol ) (4.35) 

It is customary to introduce certain macroscopic quantities, namely, the 
electric field E, the polarization P (electric dipole moment per unit volume), 



Fig. 4.2 Origin of polarization-charge density. 

Because of spatial variation of polarization more 

molecular charge may leave a given small volume 

than enters it. 




108 Classical Electrodynamics 

the charge density p, and the displacement D, defined as follows : 

E = <€> 

P = ^<Pmol> 

P = N{e mol ) 

D = E + 4ttP 

If there are several different kinds of atoms or molecules in the medium 
and perhaps extra charge is added, these definitions have the obvious 
generalizations : _ 

p = 2>«<p«> 



(4.36) 



P = INM + Pe 



(4.37) 



where N { is the number of molecules of type * per unit volume, (e t ) is their 

average charge, and (p^) is their average dipole moment. p ex is the excess 

(or free) charge density. Usually the molecules are neutral, and the total 

charge density p is just the free charge density. 

With the definitions of (4.36) or (4.37), the macroscopic divergence 

equation becomes : „ _. . .,,„, 

n V • D = 4tt-/> (4.38) 

The macroscopic equivalent of the other member of the pair (4.20) can be 
obtained by taking the curl of (4.33). Obviously the result is 

V x E = (4.39) 

For macroscopic electrostatic problems in the presence of dielectrics, 
(4.38) and (4.39) replace the microscopic equations (4.20). 

The solution for the electric field (4.33) can be expressed in terms of the 
macroscopic variables as 



E(x) 



= -V [d z x'\ p(x,) + P(x') • V'( ) 

J Llx-x'l \|x-x'|/J 



(4.40) 



The second term, describing the dipole field, has already been discussed in 
Section 1.6. 



4.4 Simple Dielectrics and Boundary Conditions 

It was mentioned in the previous section that the molecular polarization 
depends on the local electric field at the molecule. In the absence of a 
field there is no average polarization.* This means that the polarization 

* Except for electrets, which have a permanent electric polarization. 



[Sect. 4.4] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 109 

P, which is in general a function of E, can be expanded as a powers series 
in the field, at least for small fields. Any component will have an expansion 

of the form: _, ~ „ , ^, F ^ . 

Pi = 2<*h e j + 2 b m E > E k + * * ' 

A priori it is not clear how important the higher terms will be in practice. 
Experimentally it is found that the polarization as a function of applied 
field looks qualitatively as shown in Fig. 4.3. At normal temperatures and 
for fields attainable in the laboratory the linear approximation is completely 
adequate. This is not surprising if it is remembered that interatomic 
electric fields are of the order of 10 9 volts/cm. Any external field causing 
polarization is only a small perturbation. For a general anisotropic 
medium (e.g., certain crystals such as calcite and quartz), there can be six 
independent elements a u . But for simple substances, called isotropic, P 
is parallel to E with a constant of proportionality % e which is independent 
of the direction of E. Then p _ g (4.41) 

The constant % e is called the electric susceptibility of the medium. We 
then find the displacement proportional to E: 

D = eE (4.42) 

where 

€ = 1 + 47T Xe (4.43) 

is the dielectric constant. 

If the dielectric is not only isotropic, but also uniform, e is independent 
of position. Then the divergence equation can be written 



V-E = — p 

€ 



(4.44) 



and all problems in that medium are reduced to those of previous chapters, 
except that the electric fields produced by given charges are reduced by a 



Fig. 4.3 Components of polariza- 
tion as a function of applied 
electric field. 




110 Classical Electrodynamics 

Region 2 




Region 1 



Fig. 4.4 

factor 1/e. The reduction can be understood in terms of a polarization of 
the atoms which produce fields in opposition to that of the given charge. 
One immediate consequence is that the capacitance of a capacitor is increased 
by a factor of e if the empty space between the electrodes is filled with a di- 
electric with dielectric constant e (true only to the extent that fringing fields 
can be neglected). 

An important consideration is the boundary conditions on the field 
quantities E and D at surfaces where the dielectric properties vary dis- 
continuously. Consider a surface S as shown in Fig. 4.4. The unit vector 
n is normal to the surface and points from region 1 with dielectric constant 
e 1 to region 2 with dielectric constant e 2 . In exactly the same manner as in 
Section 1.6 we find, by taking a Gaussian pill box with end faces in regions 
1 and 2 parallel to the surface 5*, that 

(D 2 - DO • n = 4tt<7 (4.45) 

where a is the surface-charge density (not including polarization charge). 
Similarly, by applying Stokes's theorem to V x E = 0, we find that 

(E x - E 2 ) x n = (4.46) 

These boundary conditions on the normal component of D and the 
tangential component of E replace the microscopic conditions (1.22) and 
below. The macroscopic equivalent of (1.22) can be recovered from (4.45) 
by extracting the polarization-charge density from the left-hand side. 



4.5 Boundary-Value Problems with Dielectrics 

The methods of previous chapters for the solution of electrostatic 
boundary- value problems can readily be extended to handle the presence 
of dielectrics. In this section we will treat a few examples of the various 
techniques applied to dielectric media. 



[Sect. 4.5] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 111 

To illustrate the method of images for dielectrics we consider a point 
charge q embedded in a semi-infinite dielectric €l a distance d away from a 
plane interface which separates the first medium from another semi-infinite 
dielectric e 2 . The surface may be taken as the plane z = 0, as shown in 
Fig. 4.5. We must find the appropriate solution to the equations: 

€l V • E = 4irp, z > 

€2 V.E = 0, *<0 I (4-47) 

and V x E = 0, everywhere 

subject to the boundary conditions at z = : 

[ E y J I E b 

Since V x E = everywhere, E is derivable in the usual way from a 
potential <D. In attempting to use the image method it is natural to locate 
an image charge q' at the symmetrical position A' shown in Fig. 4.6. Then 
for z > the potential at a point P described by cylindrical coordinates 
( P , <f>, z) will be \ (a a'\ <a „m 



lim 




(4.48) 



where R x = V P 2 + (d - *) 2 , U, = V P * + (d + zf. So far the pro- 
cedure is completely analogous to the problem with a conducting material 
in place of the dielectric e 2 for z < 0. But we now must specify the potential 
for z < 0. Since there are no charges in the region z < 0, it must be a 
solution of Laplace's equation without singularities in that region. Clearly 
the simplest assumption is that for z < the potential is equivalent to that 
of a charge q" at the position A of the actual charge q\ 



O = - — 



z <0 



(4.50) 




Fig. 4.5 



112 



Classical Electrodynamics 


62 


p 


Ry^ 


^^ VRi 


q *^^^ 


\ 9 


A '!c d > 


d >\ A 


i 


i 



Fig. 4.6 



Since 



while 



1(1 
1(1) 



_ d(M 


d 


z = dz\# 

-Ht) 


2^ 


_ — P 


z = 


o~ (p 2 + d 2 f A 



the boundary conditions (4.48) lead to the requirements : 

q - q' = q" 

-(q+q') = -q" 

e l e 2 

These can be solved to yield the image charges q' and q" : 



q' = ~ 



+ «i 



« 






(4.51) 



For the two cases e 2 > e x and e 2 < e x the lines of force are shown qualita- 
tively in Fig. 4.7. 

The polarization-charge density is given by — V«P. Inside either 
dielectric, P = # e E, so that — V • P = — # e V • E = 0, except at the point 
charge q. At the surface, however, % e takes a discontinuous jump, 
Ax e = (l/47r)(ei — e 2 ) as z passes through z = 0. This implies that there 
is a polarization surface-charge density on the plane 2 = 0: 

<Vi=-(P 2 -Pi)-n (4.52) 

where n is the unit normal from dielectric 1 to dielectric 2, and P 4 is the 
polarization in the dielectric / at z = 0. Since 



P,= 



4n 



V-te 1 )™ 



[Sect. 4.5] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 113 





€2>ei 



«2<«1 



Fig. 4.7 Lines of electric force for a point charge embedded in a dielectric e x near a semi- 
infinite slab of dielectric e 2 . 



it is a simple matter to show that the polarization-charge density is 



CTpol = 



q in — € i) d 

27re 1 (e 2 + e 1 )0> 2 + d 2 ) 5 



(4.53) 



In the limit e 2 > e x the dielectric e 2 behaves much like a conductor in that 
the field inside it becomes very small and the surface-charge density (4.53) 
approaches the value appropriate to a conducting surface. 

The second illustration of electrostatic problems involving dielectrics is 
that of a dielectric sphere of radius a with dielectric constant e placed in an 
initially uniform electric field which at large distances from the sphere is 
directed along the z axis and has magnitude E , as indicated in Fig. 4.8. 
Both inside and outside the sphere there are no free charges. Consequently 
the problem is one of solving Laplace's equation with the proper boundary 
conditions at r = a. From the axial symmetry of the geometry we can 




Fig. 4.8 



H4 Classical Electrodynamics 

take the solution to be of the form: 



inside: 



®in = lA l rip i (cose) 



1=0 



outside: OTt = 2 [B t r l + Cy-^+^cos 0) 



1 = 



(4.54) 
(4.55) 



From the boundary condition at infinity (O -> — E z = —E r cos 0) we 
find that the only nonvanishing B % is B ± = -E . The other coefficients are 
determined from the boundary conditions at r = a: 



1 dO 
tangential E: 



NORMAL D: 



a dd 



1 ao 



out 



dr 



a dd 
50 mit 



dr 



(4.56) 



The first boundary condition leads to the relations : 



A x = -E + ^ 
a 6 



1 „2l+l> 



for / ^ 1 



(4.57) 



while the second gives : 



J 



^ 1= -£ -2^ 



el A, = -(/ + 1) 



C, 



for / ^ 1 



(4.58) 



The second equations in (4.57) and (4.58) can be satisfied simultaneously 
only with A l = C t = for all / ^ 1. The remaining coefficients are given 
in terms of the applied electric field E : 






(4.59) 



The potential is therefore 

<E>in = - \—^—)E r cos 6 



O ut = — E r cos d + 



€ + 2 



2/*° r 2 



cos 



(4.60) 



[Sect. 4.5] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 115 

The potential inside the sphere describes a constant electric field 
parallel to the applied field with magnitude 

£m = — , E < E (4.61) 

e + 2 

Outside the sphere the potential is equivalent to the applied field E plus 
the field of an electric dipole at the origin with dipole moment: 



■ fcriW 



(4.62) 



oriented in the direction of the applied field. The dipole moment can be 
interpreted as the volume integral of the polarization P. The polarization 
is 

P= (i^i) E = l(i^i) Eo ( 4 .63) 

\ 477 / 47T\€ + 2/ 

It is constant throughout the volume of the sphere and has a volume 
integral given by (4.62). The polarization surface-charge density is, 
according to (4.52), <r pol = (P • r)/r: 



Opol 



-lrr\7V~2) E0 



cos (4.64) 



This can be thought of as producing an internal field directed oppositely 
to the applied field, so reducing the field inside the sphere to its value (4.61), 
as sketched in Fig. 4.9. 

The problem of a spherical cavity of radius a in a dielectric medium with 
dielectric constant e and with an applied electric field E parallel to the z 
axis, as shown in Fig. 4.10, can be handled in exactly the same way as the 
dielectric sphere. In fact, inspection of boundary conditions (4.56) shows 
that the results for the cavity can be obtained from those of the sphere by 
the replacement e -> (1/e). Thus, for example, the field inside the cavity 



E 



Eq 



,-< /+ 



Fig. 4.9 Dielectric sphere in a uniform field E , showing the polarization on the left 
and the polarization charge with its associated, opposing, electric field on the right. 



116 



Classical Electrodynamics 




Fig. 4.10. Spherical cavity in a 
dielectric with a uniform field 
applied. 



is uniform, parallel to E , and of magnitude : 

3e 



E\n = 



2e+ 1 



E > E 



(4.65) 



Similarly, the field outside is the applied field plus that of a dipole at the 
origin oriented oppositely to the applied field and with dipole moment: 



'■ferrK 



(4.66) 



4.6 Molecular Polarizability and Electric Susceptibility 

In this section and the next we will consider the relation between 
molecular properties and the macroscopically defined parameter, the 
electric susceptibility Xe - Our discussion will be in terms of simple 
classical models of the molecular properties, although a proper treatment 
necessarily would involve quantum-mechanical considerations. Fortu- 
nately, the simpler properties of dielectrics are amenable to classical 
analysis. 

Before examining how the detailed properties of the molecules are related 
to the susceptibility we must make a distinction between the fields acting 
on the molecules in the medium and the external field. The susceptibility 
is defined through the relation P = Xe E, where E is the macroscopic 
electric field. In rarefied media where molecular separations are large 
there is little difference between the macroscopic field and that acting on 
any molecule or group of molecules. But in dense media with closely 
packed molecules the polarization of neighboring molecules gives rise to 
an internal field E, at any given molecule in addition to the average 
macroscopic field E, so that the total field at the molecule is E + E,. The 
internal field can be written as 



E, 



-(t+-> 



(4.67) 



[Sect. 4.6] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 117 

where sF is the contribution of molecules close to the given molecule, and 
(4tt/3)P is the contribution of the more distant molecules. It is customary 
to consider the two parts separately by imagining a spherical surface of size 
large microscopically but small macroscopically surrounding a molecule, 
as shown in Fig. 4.11, and determining the field at the center due to the 
polarization of the molecules exterior to the sphere and the resulting charge 
density induced on the surface of the sphere. This charge density is 
— P . n, where n is the outward normal from the spherical surface. The 
resulting field at the center is obviously parallel to P and has the magnitude : 

E a> f r2 da (-^ cos ex-cos o) = to p (4 68) 

J sphere T * 

giving the first term in (4.67). 

The field sP due to the molecules near by is more difficult to determine. 
Lorentz (p. 138) showed that for atoms in a simple cubic lattice s = at 
any lattice site. The argument depends on the symmetry of the problem, 
as can be seen as follows. Suppose that inside the sphere we have a cubic 
array of dipoles such as are shown in Fig. 4.12, with all their moments 
constant in magnitude and oriented along the same direction (remember 
that the sphere is macroscopically small). The positions of the dipoles are 
given by the coordinates x ijk with the components along the coordinate 
axes {ia, ja, ka), where a is the lattice spacing, and i,j, k each take on 
positive and negative integer values. The field at the origin due to all the 
dipoles is, according to (4.13), 



E = 



2 3(P • x-nk) x ijk — x ijkV (4.69) 

i,i,k 



x ijk 



The x component of the field can be written in the form : : 

P ^ 3(i 2 Pl + Up* + ikp z ) - (i 2 + f + fc 2 )Pi (470) 

ilk 



Spherical ^""" ~~^ N s\ti 
surface ~^V V 



Fig. 4.11 Calculation of the internal field — 
contribution from distant molecules. 



I — -» — - 

I ^ I p 

\ Molecule ,' 

-V- V ^ 



118 



Classical Electrodynamics 




Fig. 4.12 Calculation of the in- 
ternal field — contribution from near- 
by molecules in a simple cubic lattice. 

Since the indices run equally over positive and negative values, the cross 
terms involving (ifp 2 + ikp 3 ) vanish. By symmetry the sums: 

y i2 V f y k 2 

£ (* 2 + f + k*t £ (,- + f + k*f* £ 0* + f + k*t 

are all equal. Consequently 

E = V [3i 2 -Q- 2 + j 2 + /c 2 )] Pl _ 

1 / . a, .9. . .9. . , o^te — " 



it «V +/ + **) 



2\^ 



(4.71) 



ijk 



Similar arguments show that the y and z components vanish also. Hence 
s = for a simple cubic lattice. 

If s = for a highly symmetric situation, it seems plausible that s = 
also for completely random situations. Hence we expect amorphous 
substances like glass to have no internal field due to near-by molecules. 
Although calculations taking into account the structural details of the 
substance are necessary to obtain an accurate answer, it is a good working 
assumption that s ~ for almost all materials. 

The polarization vector P was defined in (4.36) as 

P = ^<Pmol> 

where (p mol > is the average dipole moment of the molecules. This dipole 
moment is approximately proportional to the electric field acting on the 
molecule. To exhibit this dependence on electric field we define the mole- 
cular polarizability y mol as the ratio of the average molecular dipole 
moment to the applied field at the molecule. Taking account of the internal 
field (4.67), this gives : 

<Pmol> = 7mol(E + E,) (4.72) 



>mol 



is, in principle, a function of the electric field, but for a wide range of 



[Sect. 4.7] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 119 

field strengths is a constant which characterizes the response of the 
molecules to an applied field (see Section 4.4). Equation (4.72) can be 
combined with (4.36) and (4.67) to yield: 

P = N)w(E + yP) (4.73) 

where we have assumed 5 = 0. Solving for P in terms of E and using the 
fact that P = % e E defines the electric susceptibility of a substance, we find 

X. = N I m ° l ( 4 - 74 > 

47T 

1 Ny m0 \ 

3 

as the relation between susceptibility (the macroscopic parameter) and 
molecular polarizability (the microscopic parameter). Since the dielectric 
constant is e = 1 + 4tt^ 6 , it can be expressed in terms of y mol , or 
alternatively the molecular polarizability can be expressed in terms of the 
dielectric constant : 

7mol = -^-(^) (4.75) 

4ttN\€ + 2/ 

This is called the Clausius-Mossotti equation, since Mossotti (in 1850) and 
Clausius independently (in 1879) established that for any given substance 
( e _ i)/( e + 2) should be proportional to the density of the substance.* 
The relation holds best for dilute substances such as gases. For liquids 
and solids, (4.75) is only approximately valid, especially if the dielectric 
constant is large. The interested reader can refer to the books by Bottcher, 
Debye, and Frohlich for further details. 



4.7 Models for the Molecular Polarizability 

The polarization of a collection of atoms or molecules arises in two ways : 

(a) the applied field distorts the charge distributions and so produces 
an induced dipole moment in each molecule ; 

(b) the applied field tends to line up the initially randomly oriented 
permanent dipole moments of the molecules. 

To estimate the induced moments we will consider a simple model of 

* At optical frequencies, e = « 2 , where n is the index of refraction. With n 2 replacing 
e in (4.75), the equation is sometimes called the Lorentz-Lorenz equation (1880). 



120 Classical Electrodynamics 

harmonically bound charges (electrons). Each charge e is bound under 
the action of a restoring force 

F = -ma) 2 x (4.76) 

where m is the mass of the charge, and o the frequency of oscillation 
about equilibrium. Under the action of an electric field E the charge is 
displaced from its equilibrium by an amount x given by 

mco Q 2 x = eE 

Consequently the induced dipole moment is 

e 2 

Pmoi = ex = E (4.77) 



ma>. 



This means that the polarizability is y = e 2 {mco 2 . If there are Z electrons 
per molecule,^ having a restoring force constant mco? QT/} = Z), then the 
molecular polarizability due to the electrons is : j 

Y*= L y— % (4.78) 

i 

To get a feeling for the order of magnitude of y el we can make two 
different estimates. Since y has the dimensions of a volume, its magnitude 
must be of the order of molecular dimensions or less, namely y el < 10~ 23 
cm 3 . Alternatively, we note that the binding frequencies of electrons in 
atoms must be of the order of light frequencies. Taking a typical wave- 
length of light as 3000 angstroms, we find co ~ 6 x 10 15 sec -1 . Then 
Yei ^ (e 2 /ma> 2 ) ~6x 10~ 24 cm 3 , consistent with the molecular volume 
estimate. For gases at NTP the number of molecules per cubic centimeter 
is TV = 2.7 x 10 19 , so that their susceptibilities should be of the order of 
X e < 10 -4 . This means dielectric constants differing from unity by a few 
parts in 10 3 , or less. Experimentally, typical values of dielectric constant 
are 1.00054 for air, 1.0072 for ammonia vapor, 1.0057 for methyl alcohol, 
1.000068 for helium. For solid or liquid dielectrics, N~ 10 22 - 10 23 
molecules/cm 3 . Consequently, the susceptibility can be of the order of 
unity (to within a factor 10 ±r ) as is observed.* 

The possibility that thermal agitation of the molecules could modify the 
result (4.78) for the induced dipole polarizability needs consideration. In 
statistical mechanics the probability distribution of particles in phase 

* See Handbook of Chemistry and Physics, Chemical Rubber Publishing Co., or 
American Institute of Physics Handbook, McGraw-Hill, New York, (1957), for tables of 
dielectric constants of various substances. 



[Sect. 4.7] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 111 
space (p, q space) is proportional to the Boltzmann factor 

expi-H/kT) (4.79) 

where H is the Hamiltonian. In the simple problem of a harmonically 
bound electron with an applied field in the z direction, the Hamiltonian is 

H = — p 2 + - eo 2 x 2 - eEz (4.80) 

2m 2 

where here p is the momentum of the electron. The average value of the 
dipole moment is 

(d 3 p(d 3 x(ez)exp(-HlkT) 

<*w> = ^-j (4.81) 

d*p d*x exp (-H/kT) 

The integration over (d z p) and (dx dy) can be done immediately to yield 

(pmol) = 



e\dzz exp 


r_ i /W z2 eE Y] 

L kT\ 2 /J 


dz exp 


1 |W 2 2 _ eE X 

. kT\ 2 /J 



An integration by parts in the numerator yields the result : 



e 2 



(pmoi) = ; E 

ma) Q 

the same as was found in (4.77) by elementary means, ignoring thermal 
motion. Thus the molecular polarizability (4.78) holds even in the presence 
of thermal motion. 

The second type of polarizability is that caused by the partial orientation 
of randomly oriented permanent dipole moments. This orientation polari- 
zation is important in "polar" substances such as HC1 and H 2 and was 
first discussed by Debye (1912). All molecules are assumed to possess a 
permanent dipole moment p which can be oriented in any direction in 
space. In the absence of a field thermal agitation keeps the molecules 
randomly oriented so that there is no net dipole moment. With an applied 
field there is a tendency to line up along the field in the configuration of 
lowest energy. Consequently there will be an average dipole moment. To 
calculate this we note that the Hamiltonian of the molecule is given by 

H = H - p • E (4.82) 



122 Classical Electrodynamics 




' -Nonpolar 

Fig. 4.13. Variation of molec- 
ular polarizability y m oi with 
temperature for polar and non- 
polar substances. y m oi is plot- 
\IT ■>■ ted versus T _1 . 

where H Q is a function of only the "internal" coordinates of the molecule. 
Using the Boltzmann factor (4.79), we can write the average dipole 
moment as : 

[dnp cos6ttp(*£™l\ 

<Pmoi> = -j— ^— (4.83) 

where we have chosen E along the z axis, integrated out all the irrelevant 
variables, and noted that only the component of p parallel to the field is 
different from zero. In general, (p E/kT) is very small compared to unity, 
except at low temperatures. Hence we can expand the exponentials and 
obtain the result: 

<Pna)^~ E ( 4 - 84 ) 

3 kT 

We note that the orientation polarization depends inversely on the tempera- 
ture, as might be expected of an effect in which the applied field must 
overcome the opposition of thermal agitation. 

In general both types of polarization, induced (electronic) and orienta- 
tion, are present, and the general form of the molecular polarization is 

ymoi ~ yei + - t- (4.85) 

3 kT 

This shows a temperature dependence of the form (a + b/T) so that the 
two types of polarization can be separated experimentally, as indicated in 
Fig. 4.13. For "polar" molecules, such as HC1 and H 2 0, the observed 
permanent dipole moments are of the order of an electronic charge times 
10~ 8 cm, in accordance with molecular dimensions. 



[Sect. 4.8] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 123 

4.8 Electrostatic Energy in Dielectric Media 

In Section 1.11 we discussed the energy of a system of charges in free 
space. The result 

^=iL(x)(D(x)d 3 z (4.86) 

for the energy due to a charge density p(x) and a potential O(x) cannot in 
general be taken over as it stands in our macroscopic description of 
dielectric media. The reason becomes clear when we recall how (4.86) was 
obtained. We thought of the final configuration of charge as being 
created by assembling bit by bit the elemental charges, bringing each one 
in from infinitely far away against the action of the then existing electric 
field. The total work done was given by (4.86). With dielectric media 
work is done not only to bring real (macroscopic) charge into position, 
but also to produce a certain state of polarization in the medium. If p 
and 3> in (4.86) represent macroscopic variables, it is certainly not evident 
that (4.86) represents the total work, including that done on the dielectric. 
In order to be general in our description of dielectrics we will not 
initially make any assumptions about linearity, uniformity, etc., of the 
response of a dielectric to an applied field. Rather, let us consider a small 
change in the energy bW due to some sort of change bp in the charge 
density /> existing in all space. The work done to accomplish this change 



is 



dW = (dp(x)<I>(x)d z x (4.87) 

where O(x) is the potential due to the charge density p(x) already present. 
Since V • D = Anp, we can relate the change dp to a change in the dis- 
placement of <5D : 

<5 P = — V-05D) (4.88) 

477 

Then the energy change bW can be cast into the form: 

bW = — f E • dD d*x (4.89) 

4tt J 

where we have used E = — V® and have assumed that p(x) was a localized 
charge distribution. The total electrostatic energy can now be written 
down formally, at least, by allowing D to be brought from an initial value 
D = to its final value D : 

W=— (d 3 x( D E-bD (4.90) 

477 J -JO 



124 Classical Electrodynamics 

If the medium is linear, then 

E • (5D = i(5(E • D) (4.91) 

and the total electrostatic energy is 



877 J 



W=— E.Dft (4.92) 

877 J 

This last result can be transformed into (4.86) by using E = — VO and 
V • D = A-np, or by going back to (4.87) and assuming that p and O are 
connected linearly. Thus we see that (4.86) is valid macroscopically only 
if the behavior is linear. Otherwise the energy of a final configuration must 
be calculated from (4.90) and might conceivably depend on the past 
history of the system (hysteresis effects). 

A problem of considerable interest is the change in energy when a 
dielectric object is placed in an electric field whose sources are fixed. 
Suppose that initially the eJectric field E due to a certain distribution of 
charges p (x) exists in a medium of dielectric constant e which may be a 
function of position. The initial electrostatic energy is 



^ = ^- fE .D d 3 

077 J 



where D = e E . Then with the sources fixed in position a dielectric 
object of volume V 1 is introduced into the field, changing the field from E 
to E. The presence of the object can be described by a dielectric constant 
e(x), which has the value e x inside V 1 and e outside V x . To avoid mathe- 
matical difficulties we can imagine e(x) to be a smoothly varying function 
of position which falls rapidly but continuously from e x to e at the edge 
of the volume V x . The energy now has the value 

W x = — E • D d 3 x 

Sir J 

where D = eE. The difference in the energy can be written : 
W=±- f(E.D-E .D )^ 

077 J 

1 f 

.93) 



= ^- f(E.D -D.E )^ + ^- f(E + E ) • (D - D ) d*x (4. 



The second integral can be shown to vanish by the following argument. 
Since V x (E + E ) = 0, we can write 

E + E = -V0> 



[Sect. 4.8] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 125 
Then the second integral becomes : 

j = _ J_ f VO • (D - D ) d 3 x 

877 J 

Integration by parts transforms this into 

I = — I OV • (D - D ) d 3 x = 

since V • (D — D ) = because the source charge density /> (x) is assumed 
unaltered by the insertion of the dielectric object. Consequently the energy 
change is 

W = — f(E • D - D • E ) d z x (4.94) 

877- J 

The integration appears to be over all space, but is actually only over the 
volume V x of the object, since, outside V lt D = e E. Therefore we can 
write 

W = - — f (e, - € ft )E • E d 3 x (4.95) 



= __L f ( ei _ €o )E.E ^ 

87T JVi 



If the medium surrounding the dielectric body is free space, then e = 1. 
Using the definition of polarization P, (4.95) can be expressed in the form: 

W= -|f P-E d 3 x (4.96) 

where P is the polarization of the dielectric. This shows that the energy 
density of a dielectric placed in a field E whose sources are fixed is given 
by 

w=-|P-E (4.97) 

This result is analogous to the dipole term in the energy (4.17) of a charge 
distribution in an external field. The factor \ is due to the fact that (4.97) 
represents the energy density of a polarizable dielectric in an external field, 
rather than a permanent dipole. It is the same factor \ which appears in 
(4.91). 

Equations (4.95) and (4.96) show that a dielectric body will tend to 
move towards regions of increasing field E provided e x > e . To calculate 
the force acting we can imagine a small generalized displacement of the 
body <5f . Then there will be a change in the energy bW. Since the charges 
are held fixed, there is no external source of energy and the change in field 



126 Classical Electrodynamics 

energy must be compensated for by a change in the mechanical energy of 
the body. This means that there is a force acting on the body : 

where the subscript Q has been placed on the partial derivative to indicate 
that the sources of the field are kept fixed. 

In practical situations involving the motion of dielectrics the electric 
fields are often produced by a configuration of electrodes held at fixed 
potentials by connection to an external source such as a battery. As the 
distribution of dielectric varies, charge will flow to or from the battery to 
the electrodes in order to maintain the potentials constant. This means that 
energy is being supplied from the external source, and it is of interest to 
compare the energy supplied in that way with the energy change found 
above for fixed sources of the field. We will treat only linear media so that 
(4.86) is valid. It is sufficient to consider small changes in an already 
existing configuration. From (4.86) it is evident that the change in energy 
accompanying the changes dp(x) and d<D(x) in charge density and potential 
is 

6W=U |><5(D + #<fy] d 3 x (4.99) 

Comparison with (4.87) shows that, if the dielectric properties are not 
changed, the two terms in (4.99) are equal. If, however, the dielectric 
properties are altered, 

e(x) -* e(x) + Se(x) (4.100) 

the contributions in (4.99) are not necessarily the same. In fact, we have 
just calculated the change in energy brought about by introducing a 
dielectric body into an electric field whose sources were fixed (dp = 0). 
The reason for this difference is the existence of the polarization charge. 
The change in dielectric properties implied by (4. 100) can be thought of as a 
change in the polarization-charge density. If then (4.99) is interpreted as an 
integral over both free and polarization-charge densities (i.e., a micro- 
scopic equation), the two contributions are always equal. However, it is 
often convenient to deal with macroscopic quantities. Then the equality 
holds only if the dielectric properties are unchanged. 

The process of altering the dielectric properties in some way (by moving 
the dielectric bodies, by changing their susceptibilities, etc.) in the presence 
of electrodes at fixed potentials can be viewed as taking place in two steps. 
In the first step the electrodes are disconnected from the batteries and the 



[Sect. 4.8] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 127 

charges on them held fixed (dp = 0). With the change (4.100) in dielectric 
properties, the energy change is 

dW 1 = \\pb<S> 1 d z x (4.101) 

where dd> 1 is the change in potential produced. This can be shown to 
yield the result (4.95). In the second step the batteries are connected again 
to the electrodes to restore their potentials to the original values. There 
will be a flow of charge dp 2 from the batteries accompanying the change in 
potential* M> 2 = — 6^. Therefore the energy change in the second step 



is 



dW 2 = \ f(/x50 2 + <t>dp 2 ) d 3 x = -26W 1 (4.102) 

since the two contributions are equal. In the second step we find the 
external sources changing the energy in the opposite sense and by twice 
the amount of the initial step. Consequently the net change is 

6W= -Upd^dPx (4.103) 

Symbolically 

bW v = -6W Q (4.104) 

where the subscript denotes the quantity held fixed. If a dielectric with 
€ > 1 moves into a region of greater field strength, the energy increases 
instead of decreases. For a generalized displacement d£ the mechanical 
force acting is now 

F * = + (fi (4105) 



REFERENCES AND SUGGESTED READING 

The derivation of the macroscopic equations of electrostatics by averaging over 
aggregates of atoms is presented by 

Rosenfeld, Chapter II, 

Mason and Weaver, Chapter I, Part III, 

Van Vleck, Chapter 1. 
Rosenfeld also treats the classical electron theory of dielectrics. Van Vleck's book is 
devoted to electric and magnetic susceptibilities. Specific works on electric polarization 
phenomena are those of 

Bottcher, 

Debye, 

Frohlich. 

* Note that it is necessary merely to know that M> 2 = — <5®i on the electrodes, since 
that is the only place where free charge resides. 



128 



Classical Electrodynamics 



Boundary-value problems with dielectrics are discussed in all the references on 
electrostatics in Chapters 2 and 3. 

Our treatment of forces and energy with dielectric media is brief. More extensive 
discussions, including forces on liquid and solid dielectrics, the electric stress tensor, 
electrostriction, and thermodynamic effects, may be found in 

Abraham and Becker, Band 1, Chapter V, 

Durand, Chapters VI and VII, 

Landau and Lifshitz, Electrodynamics of Continuous Media, 

Maxwell, Vol. 1, Chapter V, 

Panofsky and Phillips, Chapter 6, 

Stratton, Chapter II. 



PROBLEMS 

4.1 Calculate the multipole moments q lm of the charge distributions shown 
below. Try to obtain results for the nonvanishing moments valid for all /, but 
in each case find the first two sets of nonvanishing moments at the very least. 




(a) 




Total 

charge 

<? 




Conducting circular 
disc of radius 



{d) For the charge distribution (Jo) write down the multipole expansion 
for the potential. Keeping only the lowest-order term in the expansion, plot 
the potential in the x-y plane as a function of distance from the origin for 
distances greater than a. 

(e) Calculate directly from Coulomb's law the exact potential for (Jb) in the 
x-y plane. Plot it as a function of distance and compare with the result found 
in (d). 

Divide out the asymptotic form in parts (d) and (e) in order to see the 
behavior at large distances more clearly. 
4.2 A nucleus with quadrupole moment Q finds itself in a cylindrically symmetric 
electric field with a gradient (dEjdz) along the z axis at the position of the 
nucleus. 

(a) Show that the energy of quadrupole interaction is 



—HZ). 



(b) If it is known that Q = 2 x 10~ 24 cm 2 and that WJh is 10 Mc/sec, 
where h is Planck's constant, calculate (dE z /dz) in units of e/a 3 , where 
a = h 2 /me 2 = 0.529 x 10~ 8 cm is the Bohr radius in hydrogen. 



[Probs. 4] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 129 

(c) Nuclear-charge distributions can be approximated by a constant 
charge density throughout a spheroidal volume of semimajor axis a and 
semiminor axis b. Calculate the quadrupole moment of such a nucleus, 
assuming that the total charge is Ze. Given that Eu 153 (Z = 63) has a 
quadrupole moment Q = 2.5 x 10 -24 cm 2 and a mean radius 

R = (a + b)\2 = 7 x 10- 13 cm, 

determine the fractional difference in radius (a — b)lR. 

4.3 A localized distribution of charge has a charge density 

/>(r) = Jr- r 2 e~ r sin 2 

647T 

(a) Make a multipole expansion of the potential due to this charge 
density and determine all the nonvanishing multipole moments. Write 
down the potential at large distances as a finite expansion in Legendre 
polynomials. 

(b) Determine the potential explicitly at any point in space, and show 
that near the origin 

(c) If there exists at the origin a nucleus with a quadrupole moment 
q = io~ 24 cm 2 , determine the magnitude of the interaction energy, assuming 
that the unit of charge in P (r) above is the electronic charge and the unit of 
length is the hydrogen Bohr radius a = h 2 lme* = 0.529 x lO" 8 cm. 
Express your answer as a frequency by dividing by Planck's constant h. 

The charge density in this problem is that for the m = ±1 states of the 
2/7 level in hydrogen, while the quadrupole interaction is of the same order 
as found in molecules. 

4.4 A very long, right circular, cylindrical shell of dielectric constant e and inner 
and outer radii a and b, respectively, is placed in a previously uniform 
electric field E with its axis perpendicular to the field. The medium inside 
and outside the cylinder has a dielectric constant of unity. 

(a) Determine the potential and electric field in the three regions, 
neglecting end effects. 

(6) Sketch the lines of force for a typical case of b ~ 2a. 

(c) Discuss the limiting forms of your solution appropriate for a solid 
dielectric cylinder in a uniform field, and a cylindrical cavity in a uniform 
dielectric. 

4.5 A point charge q is located in free space a distance d from the center of a 
dielectric sphere of radius a {a < d) and dielectric constant e. 

(a) Find the potential at all points in space as an expansion in spherical 
harmonics. 

(Jb) Calculate the rectangular components of the electric field near the 
center of the sphere. 

(c) Verify that, in the limit e -► oo, your result is the same as that for the 
conducting sphere. 



130 Classical Electrodynamics 

4.6 Two concentric conducting spheres of inner and outer radii a and b, 
respectively, carry charges ± Q. The empty space between the spheres is 
half-filled by a hemispherical shell of dielectric (of dielectric constant e), as 
shown in the figure. 




(a) Find the electric field everywhere between the spheres. 

(b) Calculate the surface-charge distribution on the inner sphere. 

(c) Calculate the polarization-charge density induced on the surface of the 
dielectric at r = a. 

4.7 The following data on the variation of dielectric constant with pressure are 
taken from the Smithsonian Physical Tables, 9th ed., p. 424: 



Air at 292°K 



Pressure (atm) 



20 1.0108 Relative density of 

40 1.0218 air as a function of 



60 1.0333 



pressure is given in 



80 1.0439 AIP Handbook, p. 

100 1.0548 4-83. 

Pentane (C 5 H 12 ) at 303 °K 
Pressure (atm) Density (gm/cm 3 ) e 



1 


0.613 


1.82 


10 3 


0.701 


1.96 


x 10 3 


0.796 


2.12 


x 10 3 


0.865 


2.24 


x 10 3 


0.907 


2.33 



12 

Test the Clausius-Mossotti relation between dielectric constant and density 
for air and pentane in the ranges tabulated. Does it hold exactly ? Approxi- 
mately? If approximately, discuss fractional variations in density and 
(e — 1). For pentane, compare the Clausius-Mossotti relation to the cruder 
relation, (e — 1) <x density. 

4.8 Water vapor is a polar gas whose dielectric constant exhibits an appreciable 
temperature dependence. The following table gives experimental data on 
this effect. Assuming that water vapor obeys the ideal gas law, calculate the 
molecular polarizability as a function of inverse temperature and plot it. 
From the slope of the curve, deduce a value for the permanent dipole 



[Probs. 4] Multipoles, Electrostatics of Macroscopic Media, Dielectrics 131 

moment of the H 2 molecule (express the dipole moment in esu— stat- 
coulomb-centimeters) . 



(°K) 


Pressure (cm Hg) 


(e - 1) x 10 5 


393 


56.49 


400.2 


423 


60.93 


371.7 


453 


65.34 


348.8 


483 


69.75 


328.7 



4.9 Two long, coaxial, cylindrical conducting surfaces of radii a and b are 
lowered vertically into a liquid dielectric. If the liquid rises a distance h 
between the electrodes when a potential difference V is established between 
them, show that the susceptibility of the liquid is 



X e - 



(b 2 - a 2 ) P gh In (bid) 



V 2 



where p is the density of the liquid, g is the acceleration due to gravity, and 
the susceptibility of air is neglected. 



5 



Magnetostatics 



5.1 Introduction and Definitions 

In the preceding chapters various aspects of electrostatics (i.e., the 
fields and interactions of stationary charges and boundaries) have been 
studied. We now turn to steady-state magnetic phenomena. From an 
historical point of view, magnetic phenomena have been known and 
studied for at least as long as electric phenomena. Lodestones were known 
in ancient times ; the mariner's compass is a very old invention ; Gilbert's 
researches on the earth as a giant magnet date from before 1600. In 
contrast to electrostatics, the basic laws of magnetic fields did not follow 
straightforwardly from man's earliest contact with magnetic materials. 
The reasons are several, but they all stem from the radical difference 
between magnetostatics and electrostatics : there are no free magnetic 
charges. This means that magnetic phenomena are quite different from 
electric phenomena and that for a long time no connection was established 
between them. The basic entity in magnetic studies was what we now know 
as a magnetic dipole. In the presence of magnetic materials the dipole 
tends to align itself in a certain direction. That direction is by definition 
the direction of the magnetic-flux density, denoted by B, provided the 
dipole is sufficiently small and weak that it does not perturb the existing 
field. The magnitude of the flux density can be defined by the mechanical 
torque N exerted on the magnetic dipole : 

N = |jl x B (5.1) 

where {a is the magnetic moment of the dipole, defined in some suitable 
set of units. * 

* In analogy with the 100 strokes of cat's fur on an amber rod, we might define our unit 
of dipole strength as that of a i-inch finishing nail which has been stroked slowly 100 
times with a certain "standard" lodestone held in a certain standard orientation. With 
a little thought we might even think of a more reliable and reproducible standard! 

132 



[Sect. 5.2] Magnetostatics 133 

Already, in the definition of the magnetic-flux density B (sometimes 
called the magnetic induction), we have a more complicated situation than 
for the electric field. Further quantitative elucidation of magnetic 
phenomena did not occur until the connection between currents and 
magnetic fields was established. A current corresponds to charges in 
motion and is described by a current density J, measured in units of positive 
charge crossing unit area per unit time, the direction of motion of the 
charges defining the direction of J. In electrostatic units, current density 
is measured in statcoulombs per square centimeter-second, and is some- 
times called statamperes per square centimeter, while in mks units it is 
measured in coulombs per square meter-second or amperes per square 
meter. If the current density is confined to wires of small cross section, 
we usually integrate over the cross-sectional area and speak of a current 
of so many statamperes or amperes flowing along the wire. 

Conservation of charge demands that the charge density at any point 
in space be related to the current density in that neighborhood by a 
continuity equation: 

d -P + V • J = (5.2) 

dt 

This expresses the physical fact that a decrease in charge inside a small 
volume with time must correspond to a flow of charge out through the 
surface of the small volume, since the total number of charges must be 
conserved. Steady-state magnetic phenomena are characterized by no 
change in the net charge density anywhere in space. Consequently in 
magnetostatics 

V • J = (5.3) 

We now proceed to discuss the experimental connection between current 
and magnetic-flux density and to establish the basic laws of magneto- 
statics. 



5.2 Biot and Savart Law 

In 1819 Oersted observed that wires carrying electric currents produced 
deflections of permanent magnetic dipoles placed in their neighborhood. 
Thus the currents were sources of magnetic-flux density. Biot and Savart 
(1820), first, and Ampere (1820-1825), in much more elaborate and 
thorough experiments, established the basic experimental laws relating the 
magnetic induction B to the currents and established the law of force 
between one current and another. Although not in the form in which 



134 



Classical Electrodynamics 




Fig. 5.1 Elemental magnetic induction 
dB due to current element I d\. 

Ampere deduced it, the basic relation is the following. If d\ is an element 
of length (pointing in the direction of current flow) of a filamentary wire 
which carries a current / and x is the coordinate vector from the element 
of length to an observation point P, as shown in Fig. 5.1, then the 
elemental flux density dB at the point P is given in magnitude and direction 
by 

(d\ x x) 



dB = kl 



|x| 



(5.4) 



It should be noted that (5.4) is an inverse square law, just as is Coulomb's 
law of electrostatics. However, the vector character is very different. 

If, instead of a current flowing there is a single charge q moving with a 
velocity v, then the flux density will be* 



B = kq — = kv x E 



(5.5) 



where E is the electrostatic field of the charge q. (This flux density is, 
however, time varying. We shall restrict the discussions in the present 
chapter to steady-state current flow.) 

In (5.4) and (5.5) the constant k depends on the system of units used, as 
discussed in detail in the Appendix. If current is measured in esu, but the 
flux density is measured in emu, the constant is k = 1/c, where c is found 
experimentally to be equal to the velocity of light in vacuo (c = 2.998 x 
10 10 cm/sec). This system of units is called the Gaussian system. To insert 
the velocity of light into our equations at this stage seems a little artificial, 
but it has the advantage of measuring charge and current in a consistent 
set of units so that the continuity equation (5.2) retains its simple form, 
without factors of c. We will adopt the Gaussian system here. 

Assuming that linear superposition holds, the basic law (5.4) can be 
integrated to determine the magnetic-flux density due to various config- 
urations of current-carrying wires. For example, the magnetic induction 

* True only for particles moving with velocities small compared to that of light. 



[Sect. 5.2] 



Magnetostatics 



135 



B of the long straight wire shown in Fig. 5.2 carrying a current / can be 
seen to be directed along the normal to the plane containing the wire and 
the observation point, so that the lines of magnetic induction are concentric 
circles around the wire. The magnitude of B is given by 



IBI 



-^ 



dl 



oo (r 2 + n 



2\ 3 A 



2/ 
cR 



(5.6) 



where R is the distance from the observation point to the wire. This is the 
experimental result first found by Biot and Savart and is known as the 
Biot-Savart law. Note that the magnitude of the induction B varies with 
R in the same way as the electric field due to a long line charge of uniform 
linear-charge density. This analogy shows that in some circumstances 
there may be a correspondence between electrostatic and magnetostatic 
problems, even though the vector character of the fields is different. We 
shall see more of that in later sections. 

Ampere's experiments did not deal directly with the determination of 
the relation between currents and magnetic induction, but were concerned 
rather with the force which one current-carrying wire experiences in the 
presence of another. Since we have already introduced the idea that a 
current element produces a magnetic induction, we phrase the force law as 
the force experienced by a current element I x d\ x in the presence of a 
magnetic induction B. The elemental force is 



d¥ = -1 (dl x x B) 
c 



(5.7) 



I x is the current in the element (measured in esu), B is the flux density (in 
emu), and c is the velocity of light. If the external field B is due to a closed 
current loop #2 with current 7 2 , then the total force which a closed current 




Fig. 5.2 



136 Classical Electrodynamics 




Fig. 5.3 Two Amperian current loops, 
loop #1 with current I x experiences is [from (5.4) and (5.7)]: 

F -M.4^i x ^ x ^ (5.8) 

c 2 JJ |x 12 | 3 

The line integrals are taken around the two loops; x 12 is the vector 
distance from line element d\ 2 to dl lt as shown in Fig. 5.3. This is the 
mathematical statement of Ampere's observations about forces between 
current-carrying loops. By manipulating the integrand it can be put in a 
form which is symmetric in dl x and d\ 2 and which explicitly satisfies 
Newton's third law. Thus 

d\ x x (dl 2 x x 12 ) = _ {dli . dh) _x^ + dl J d ±lM, (5 .9) 

ix 12 r |x 12 i \ |x 12 i / 

The second term involves a perfect differential in the integral over dl v 
Consequently it gives no contribution to the integral (5.8), provided the 
paths are closed or extend to infinity. Then Ampere's law of force between 
current loops becomes 

_ hh Li (<fll • ^2)Xi2 



c 2 J J |x 12 | 3 

showing symmetry in the integration, apart from the necessary vectorial 
dependence on x 12 . 

Each of two long, parallel, straight wires a distance d apart, carrying 
currents I x and I 2 , experiences a force per unit length directed perpen- 
dicularly towards the other wire and of magnitude, 

F = ^ (5.11) 

c 2 d 

The force is attractive (repulsive) if the currents flow in the same (opposite) 
directions. The forces which exist between current-carrying wires can be 



[Sect. 5.3] Magnetostatics 137 

used to define magnetic-flux density in a way that is independent of per- 
manent magnetic dipoles.* We will see later that the torque expression 
(5.1) and the force result (5.7) are intimately related. 

If a current density J(x) is in an external magnetic-flux density B(x), the 
elementary force law implies that the total force on the current distribution 
is 



-if' 



J(x) x B(x) d 3 x (5.12) 

cJ 

Similarly the total torque is 

'x x (J x B)d 3 x (5.13) 



HJ' 



These general results will be applied to localized current distributions in 
Section 5.6. 



5.3 The Differential Equations of Magnetostatics 
and Ampere's Law 

The basic law (5.4) for the magnetic induction can be written down in 
general form for a current density J(x) : 

B(x) = - f J(x') x (x ~ x '> d 3 x> (5.14) 

cJ |x — x'| 3 

This expression for B(x) is the magnetic analog of electric field in terms of 
the charge density: 

E(x) = f P (x-) [ X ~ X ^ V (5.15) 

J |x — X \ d 

Just as this result for E was not as convenient in some situations as 
differential equations, so (5.14) is not the most useful form for magneto- 
statics, even though it contains in principle a description of all the 
phenomena. 

In order to obtain the differential equations equivalent to (5.14) we 
transform (5.14) into the form: 

B(x) = - V x f J(x>) d 3 x' (5.16) 

c J |x _ x'| 

* In fact, (5.1 1) is the basis of the internationally accepted standard of current (actually 
I/c here). See the Appendix. 



138 Classical Electrodynamics 

From (5.16) it follows immediately that the divergence of B vanishes: 

V-B = (5.17) 

This is the first equation of magnetostatics and corresponds toVxE = 
in electrostatics. By analogy with electrostatics we now calculate the curl 
ofB: 

VxB = -VxVx f J(x>) dV (5.18) 

c J |x — x'l 

With the identity V x (V x A) = V(V • A) - V 2 A for an arbitrary 
vector field A, expression (5.18) can be transformed into 

V x B = I V fj(x') • v(— i— ) dV - i f J(x')V 2 (-^—) d 3 x' 
c J \|x — x'|/ c J \|x — x|/ 

(5.19) 
Using the fact that 

v(_L_) _ -W-?-) 

\|x-x'|/ \|x-x'|/ 

and 

V 2 ( ) = -47t6(x - x') 

\|x — x'|/ 

the integrals in (5.19) can be written: 

V x B = - - V f J(x') • V ( ) d z x' + — J(x) (5.20) 

c J \|x — x'|/ c 

Integration by parts yields 

v X B = ±r J + ivf v '- J(x ' ) dV (5.20 

c c J \x — x'l 

But for steady-state magnetic phenomena V • J = 0, so that we obtain 

VxB = — J (5.22) 

c 

This is the second equation of magnetostatics, corresponding to V • E = 
Airp in electrostatics. 

In electrostatics Gauss's law (1.11) is the integral form of the equation 
V • E = Airp. The integral equivalent of (5.22) is called Ampere's law. It 
is obtained by applying Stokes's theorem to the integral of the normal 



[Sect. 5.4] 



Magnetostatics 



139 




Fig. 5.4 

component of (5.22) over an open surface S bounded by a closed curve C, 
as shown in Fig. 5.4. Thus 



is transformed into 



I V x B • n </a = — f J • n 

Js c Js 

<t B • d\ = — J • n da 

Jc c Js 



da 



(5.23) 



(5.24) 



Since the surface integral of the current density is the total current /passing 
through the closed curve C, Ampere's law can be written in the form : 



B.<fl = ^I 



(5.25) 



Just as Gauss's law can be used for calculation of the electric field in highly 
symmetric situations, so Ampere's law can be employed in analogous 
circumstances. 



5.4 Vector Potential 



The basic differential laws of magnetostatics are given by 



V x 


B = 


4tt 

— J 

c 


V- 


B = 


= 



(5.26) 



The problem is how to solve them. If the current density is zero in the 
region of interest, V x B = permits the expression of the vector 
magnetic induction B as the gradient of a magnetic scalar potential, 
B = — V<I> M . Then (5.26) reduces to Laplace's equation for <b M , and all 
our techniques for handling electrostatic problems can be brought to 
bear. There are a large number of problems which fall into this class, but 
we will defer discussion of them until later in the chapter. The reason 



140 Classical Electrodynamics 

is that the boundary conditions are different from those encountered in 
electrostatics, and the problems usually involve macroscopic media 
with magnetic properties different from free space with charges and cur- 
rents. 

A general method of attack is to exploit the second equation in (5.26). 
If V • B = everywhere, B must be the curl of some vector field A(x), 
called the vector potential, 

B(x) = V x A(x) (5.27) 

We have, in fact, already written B in this form (5.16). Evidently, from 
(5.16), the general form of A is 

A(x) = - f J(x/) d z x' + VT( X ) (5.28) 

c J |x — x'| 

The added gradient of an arbitrary scalar function *F shows that, for a 
given magnetic induction B, the vector potential can be freely transformed 
according to 

A->A+VT (5.29) 

This transformation is called a gauge transformation. Such transformations 
on A are possible because (5.27) specifies only the curl of A. For a 
complete specification of a vector field it is necessary to state both its curl 
and its divergence. The freedom of gauge transformations allows us to 
make V • A have any convenient functional form we wish. 
If (5.27) is substituted into the first equation in (5.26), we find 



or 



V x (V x A) = — J 
c 



V(V • A) - V 2 A = — J 
c 



(5.30) 



If we now exploit the freedom implied by (5.29), we can make the con- 
venient choice of gauge, * V • A = 0. Then each rectangular component 
of the vector potential satisfies Poisson's equation, 

V 2 A=-— J (5.31) 

c 

* The choice is called the Coulomb gauge, for a reason which will become apparent 
only in Section 6.5. 



[Sect. 5.5] Magnetostatics 141 

From our discussions of electrostatics it is clear that the solution for A in 
unbounded space is (5.28) with T = : 

A( HI^V V (5 - 32) 

The condition W = can be understood as follows. Our choice of gauge, 
V • A = 0, reduces to V 2t F = 0, since the first term in (5.28) has zero 
divergence because of V • J = 0. If V 2 T = holds in all space, Y must 
vanish identically. 



5.5 Vector Potential and Magnetic Induction for 
a Circular Current Loop 

As an illustration of the calculation of magnetic fields from given 
current distributions we consider the problem of a circular loop of radius 
a, lying in the x-y plane, centered at the origin, and carrying a current /, as 
shown in Fig. 5.5. The current density J has only a component in the <f> 
direction, 

J^ = /<5(cos 0') d (r ' ~ a) (5.33) 

a 

The delta functions restrict current flow to a ring of radius a. Only a <f> 
component of J means that A will have only a (f> component also. But this 
component A^ cannot be calculated by merely substituting J+ into (5.32). 
Equation (5.32) holds only for rectangular components of A.* Thus we 
write rectangular components of J: 

J* = — J 4> sin 4>' 1 

(5.34) 

jy — j f COS <f> J 

Since the geometry is cylindrically symmetric, we may choose the obser- 
vation point in the x-z plane (<f> = 0) for purposes of calculation. Then it is 
clear that the x component of the vector potential vanishes, leaving only 

* The reason is that the vector Poisson's equation (5.31) can be treated as three 
uncoupled scalar equations, V 2 A ( = (—knjc)Ji, only if the components A { , J f are 
rectangular components. If A is resolved into orthogonal components with unit vectors 
which are functions of position, the differential operation involved in (5.31) mixes the 
components together, giving coupled equations. See Morse and Feshbach, pp. 51 and 
116-117. 



142 



Classical Electrodynamics 
z 




Fig. 5.5 
the y component, which is A+. Thus 



M r. 9) = i- |> dr' iff co, ^ «co. 8-) *r' - ,) (5 .35) 
* ca J |x — x | 

where |x - x'| = [r 2 + r' 2 - 2rr'(cos cos 0' + sin sin (9' cos <£')]^- 

We first consider the straightforward evaluation of (5.35). Integration 
over the delta functions leaves the result 



^(r,0) = -P- 
c Jo (a 



cos <f>' d<f>' 



+ r 2 — 2ar sin cos <£')' 



(5.36) 



This integral can be expressed in terms of the complete elliptic integrals K 
and E: 

l~ (2 - k 2 )K(k) - lEjkj 

;L /c 2 



A^r, 0) = 



41a 



cVa 2 + r 2 + 2arsin0' 
where the argument of the elliptic integrals is 

Aar sin 



(5.37) 



k 2 = 



a 2 + r 2 + 2ar sin 
The components of magnetic induction, 

1 d 1 

B r = — r-7 ^ ( sin dA +) 
rs.va.Qdd 

*•- — a^ 

#. = 



(5.38) 



[Sect. 5.5] 



Magnetostatics 



143 



can also be expressed in terms of elliptic integrals. But the results are not 
particularly illuminating (useful, however, for computation). 

For small k 2 , corresponding to a > r, a < r, or < 1, the square 
bracket in (5.37) reduces to (ttA: 2 /16). Then the vector potential becomes 
approximately 

AJr, 0) = — .- / Sin0 • ^ (5-39) 



c (a 2 + r 2 + 2ar sin 0)' 



The corresponding fields are 



B. 



lira 2 Ma 2 + 2r 2 + ar sin 0) 



cos 



(a 2 + r 2 + 2ar sin 0) 



?A 



ha 2 . a ( 2a 2 -r 2 + ar sin 0) 

B ~ sin —r : — rrr: 

e c (a 2 + r 2 + 2arsmd) A 



(5.40) 



These can easily be specialized to the three regions, near the axis (0 < 1), 
near the center of the loop (r < a), and far from the loop (r > a). 
Of particular interest are the fields far from the loop: 






cos 



sin 



(5.41) 



Comparison with the electrostatic dipole fields (4.12) shows that the 
magnetic fields far away from a circular current loop are dipole in character. 
By analogy with electrostatics we define the magnetic dipole moment of the 
loop to be 

7r/a2 (5.42) 



m = 



We will see in the next section that this is a special case of a general 
result— localized current distributions give dipole fields at large distances; 
the magnetic moment of a plane current loop is the product of the area of 
the loop times 7/c. 

Although we have obtained a complete solution to the problem in 
terms of elliptic integrals, we will illustrate the use of a spherical harmonic 
expansion to point out similarities and differences between the magneto- 
static and electrostatic problems. Thus we return to (5.35) and substitute 



144 Classical Electrodynamics 

the spherical expansion (3.70) for |x — x'| -1 : 
4rrl„ ^Y lm (6,0) 



^ = ^ReV 



V 



ca 



^-,21+1 

l,m 



X I r' 2 dr' dQ.' <5(cos 6') d(r' - a)e**' -^ Y* m (6', f ) (5.43) 
J rt 1 



The presence of e l4> ' means that only m = + 1 will contribute to the sum. 
Hence 



A.J. — 



Sir 2 Ia 



*|^£M'-?*M w 



where now r< (r^ is the smaller (larger) of a and r. The square-bracketed 
quantity is a number depending on /: 



2/ + 1 
4tt/(/ + 1) 



JV<o) = 



o, 



Then ^ can be written 



for J even 

' (-ir +1 r(n + f) l (5>45) 
4tt/(/ + i)L r(n + i)r(f) J' 

for / = In + 1 



2/+ 1 



A +- c Z 2 -(n + 1)! r 2 >" +2 F2 * +l(C ° S G) (546) 

n = 

where (2« - 1)!! = (2n - 1)(2« - 3)(- • •) x 5 X 3 x 1, and the n = 
coefficient in the sum is unity by definition. To evaluate the radial com- 
ponent of B from (5.38) we need 



f(Vl-* 2 P?(*)) = /(/+ 1)^0*0 
dx 



(5.47) 



Then we find 



_ iTTla ^ (-l)"(2n + l)H r 2 < n+1 ( m ( . 



cr *—< 

n = 



2 n n\ 



The 6 component of B is similarly 



B a = 



7r/a a ^(-l) n (2n + 1)!! 



* 2 ^ (-D w ( 
2L, 2\n 



+ 1)! 



2n + 



2n + 

7 3 W 



w 



Pj B+1 (cos 0) 

(5.49) 



[Sect. 5.6] Magnetostatics 145 

The upper line holds for r < a, and the lower line for r > a. For r > a, 
only the n = term in the series is important. Then, since P^cos 0) = 
-sin 0, (5.48) and (5.49) reduce to (5.41). For r < a, the leading term is 
again n = 0. The fields are then equivalent to a magnetic induction 
lirl/ac in the z direction, a result that can be found by elementary means. 

We note a characteristic difference between this problem and a cor- 
responding cylindrically symmetric electrostatic problem. Associated 
Legendre polynomials appear, as well as ordinary Legendre polynomials. 
This can be traced to the vector character of the current and vector 
potential, as opposed to the scalar properties of charge and electrostatic 
potential. 

Another mode of attack on the problem of the loop is to employ an 
expansion in cylindrical waves. Instead of (3.70) as a representation of 
|x — x'| _1 we may use the cylindrical form (3.148) or (3.149). The appli- 
cation of this technique to the circular loop will be left to the problems. It 
is generally useful for any current distribution which involves current 
flowing only in the <f> direction. 



5.6 Magnetic Fields of a Localized Current Distribution; 
Magnetic Moment 

We now consider the properties of a general current distribution which 
is localized in a small region of space, "small" being relative to the scale 
of length of interest to the observer. The proper treatment of this problem, 
in analogy with the electrostatic multipole expansion, demands a discussion 
of vector spherical harmonics. These are presented in Chapter 16 in 
connection with multipole radiation. We will be content here with only 
the lowest order of approximation. Starting with (5.32), we expand the 
denominator in powers of x' measured relative to a suitable origin in the 
localized current distribution, shown schematically in Fig. 5.6: 

1 = — + ^-^- + • • • (5.50) 

|x - x'| |x| jx| 3 



Then a given component of the vector potential will have the expansion, 
A&) = J- f J,(x') dV + -*- • L(x')x'dV + • • • (5.51) 

c|x|j qxr j 

For a localized steady-state current distribution the volume integral of J 
vanishes because V • J = 0. Consequently the first term, corresponding to 
the monopole term in an electrostatic expansion, vanishes. 



146 



Classical Electrodynamics 



The integrand of the second term can be manipulated into a more 
convenient form by using the triple vector product. Thus 



(x • x') J = (x • J)x' - x x (x' x J) 



(5.52) 



The volume integral of the first term on the right can be shown to be the 
negative of the integral of the left-hand side of (5.52). Thus we consider 
the integral, 

y jX ! d 3 x' = J V • (*/JK-' d z x' = - f x/(J . V>/ d 3 x' 

= -jz/Jid 3 *' (5.53) 

The step from the first integral to the second depends on V • J = 0; the 
following step involves an integration by parts. With this identity (5.52) 
can be written in integrated form as 

J (x • x')J(x') d 3 x' = -|x x J [x' x J(x')] d 3 x' (5.54) 

We now define the magnetic moment of the current distribution J as 

= ^Jx' x J(x')d 3 x' (5.55) 



m 



Note that it is sometimes useful to consider the integrand in (5.55) as a 
magnetic-moment density or magnetization. We denote the magnetization 
due to the current density J by 

Jl = ^- (x x J) (5.56) 

2c 

The vector potential (5.51) can be expressed in terms of m as 

... m x x , e e ~ 

AW = -^- (5.57) 

This is the lowest nonvanishing term in the expansion of A for a localized 
steady-state current distribution. The magnetic induction B can be 




Fig. 5.6 Localized current density 
J(x') gives rise to a magnetic induc- 
tion at the point P with coordi- 
nate x. 



[Sect. 5.6] 



Magnetostatics 



141 




Fig. 5.7 



calculated directly by evaluating the curl of (5.57): 



3„(n-m)-n, + ^ m3(x) 

x r 3 



(5.58) 



Here n is a unit vector in the direction x. Since (5.57) and (5.58) have 
meaning only outside the current distribution, we drop the delta function 
term. The magnetic induction (5.58) has exactly the form (4.13) of the 
field of a dipole. This is the generalization of the result found for the 
circular loop in the last section. Far away from any localized current 
distribution the magnetic induction is that of a magnetic dipole of dipole 
moment given by (5.55). 

If the current is confined to a plane, but otherwise arbitrary, loop, the 
magnetic moment can be expressed in a simple form. If the current / flows 
in a closed circuit whose line element is d\, (5.55) becomes 



m = — 

2c 



x d\ 



(5.59) 



For a plane loop such as that in Fig. 5.7, the magnetic moment is perpendi- 
cular to the plane of the loop. Since \{x x d\) = da, where da is the 
triangular element of the area defined by the two ends off/1 and the origin, 
the loop integral in (5.59) gives the total area of the loop. Hence the 
magnetic moment has magnitude, 

(5.60) 



|m| = - X (Area) 
c 



regardless of the shape of the circuit. 

If the current distribution is provided by a number of charged particles 
with charges q t and masses M* in motion with velocities v*, the magnetic 
moment can be expressed in terms of the orbital angular momentum of 
the particles. The current density is 

J=24iM(x- Xi ) < 5 - 61) 



148 Classical Electrodynamics 

where x,. is the position of the ith particle. Then the magnetic moment 
(5.55) becomes 

i 

The vector product (x t x v^) is proportional to the rth particle's orbital 
angular momentum, L< = M^ x v,). Thus (5.62) becomes 

m =2^ L * ( 5 - 63 ) 

^ * 

If all the particles in motion have the same charge to mass ratio (qjM t = 
e(M), the magnetic moment can be written in terms of the total orbital 
angular momentum L: 



m 



= ^-Tl,= 



2Mc^ 2Mc 



(5.64) 



This is the well-known classical connection between angular momentum 
and magnetic moment which holds for orbital motion even on the atomic 
scale. But this classical connection fails for the intrinsic moment of 
electrons and other elementary particles. For electrons, the intrinsic 
moment is slightly more than twice as large as implied by (5.64), with the 
spin angular momentum S replacing L. Thus we speak of the electron 
having a g factor of 2(1.001 17). The departure of the magnetic moment 
from its classical value has its origins in relativistic and quantum-mechanical 
effects which we cannot consider here. 



5.7 Force and Torque on a Localized Current Distribution in an External 
Magnetic Induction 

If a localized distribution of current is placed in an external magnetic 
induction B(x), it experiences forces and torques according to Ampere's 
laws. The general expressions for the total force and torque are given by 
(5.12) and (5.13). If the external magnetic induction varies slowly over 
the region of current, a Taylor's series expansion can be utilized to find 
the dominant terms in the force and torque. A component of B can be 
expanded around a suitable origin, 

B i (x) = B i (0) + x.VB i (0) + ... (5.65) 

The force (5.12) then becomes 

F = - - B(0) x J J(x') dV + if J(x') x [(x' • V)B(O)] d z x' + • • • (5.66) 



[Sect. 5.7] Magnetostatics 149 

Since the volume integral of J vanishes for steady-state currents, the 
lowest-order term is the one involving the gradient of B. Because the 
integrand involves J and x, in addition to VB, we expect that the integral 
can be somehow transformed into the magnetic moment (5.55). To 
accomplish this we use 

J x [(x' • V)B] = J x V(x' • B) = - V x [J(x' • B)] (5.67) 

The first step depends on the fact that V x B = for the external field, 
and that the gradient operator operates only on B. Then the force can be 
written 

F=--Vx f J(x' • B) d 3 x' + • • • (5.68) 

Use can now be made of identity (5.54) with the fixed vector x replaced by 
B. Then we obtain 

F = V x (B x m) = (m • V)B = V(m • B) (5.69) 

where m is the magnetic moment (5.55). The second form in (5.69) follows 
from V • B = 0, while the third depends on V x B = 0. 

A localized current distribution in a nonuniform magnetic induction 
experiences a force proportional to its magnetic moment m and given by 
(5.69). One simple application of this result is the time-average force on a 
charged particle spiraling in a nonuniform magnetic field. As is well 
known, a charged particle in a uniform magnetic induction moves in a 
circle at right angles to the field and with constant velocity parallel to the 
field, tracing out a helical path. The circular motion is, on the time average, 
equivalent to a circular loop of current which will have a magnetic moment 
given by (5.60). If the field is not uniform but has a small gradient (so that 
in one turn around the helix the particle does not feel significantly different 
field strengths), then the motion of the particle can be discussed in terms 
of the force on the equivalent magnetic moment. Consideration of the 
signs of the moment and the force shows that charged particles tend to be 
repelled by regions of high flux density, independent of the sign of their 
charge. This is the basis of the so-called "magnetic mirrors" discussed in 
Section 12.10 from another point of view. 

The total torque on the localized current distribution is found in a 
similar way by inserting expansion (5.65) into (5.13). Here the zeroth- 
order term in the expansion contributes. Keeping only this leading term, 
we have 



N 



= if*' x [J x B(0)] d*x' (5.70) 



150 Classical Electrodynamics 

Writing out the triple vector product, we get 

N = i J*[(x' • B)J - (x' • J)B] dV (5.71) 

The first integral is the same one considered in (5.68). Hence we can write 
down its value immediately. The second integral vanishes for a localized 
steady-state current distribution, as can be seen from the identity, 
V • (x 2 J) = 2(x • J) + x 2 V • J. The leading term in the torque is therefore 

N = m x B(0) (5.72) 

This is the familiar expression for the torque on a dipole, discussed in 
Section 5.1 as one of the ways of defining the magnitude and direction of 
the magnetic induction. 

The potential energy of a permanent magnetic moment (or dipole) in 
an external magnetic field can be obtained from either the force (5.69) or 
the torque (5.72). If we interpret the force as the negative gradient of a 
potential energy U, we find 

U = -m • B (5.73) 

For a magnetic moment in a uniform field the torque (5.72) can be inter- 
preted as the negative derivative of U with respect to the angle between B 
and m. This well-known result for the potential energy of a dipole shows 
that the dipole tends to orient itself parallel to the field in the position of 
lowest potential energy. 

We remark in passing that (5.73) is not the total energy of the magnetic 
moment in the external field. In bringing the dipole m into its final 
position in the field, work must be done to keep the current J which 
produces m constant. Even though the final situation is a steady-state, 
there is a transient period initially in which the relevant fields are time 
dependent. This lies outside our present considerations. Consequently 
we will leave the discussion of the energy of magnetic fields to Section 6.2, 
after having treated Faraday's law of induction. 



5.8 Macroscopic Equations 

So far we have dealt with the basic laws (5.17) and (5.22) of steady-state 
magnetic fields as microscopic equations in the sense of Chapter 4. We 
have assumed that the current density J was a completely known function 
of position. In macroscopic problems this is often not true. The atoms in 
matter have electrons which give rise to effective atomic currents the 
current density of which is a rapidly fluctuating quantity. Only its average 



[Sect. 5.8] Magnetostatics 151 

over a macroscopic volume is known or pertinent. Furthermore, the 
atomic electrons possess intrinsic magnetic moments which cannot be 
expressed in terms of a current density. These moments can give rise to 
dipole fields which vary appreciably on the atomic scale of dimensions. 

To treat these atomic contributions we proceed similarly to Section 4.3. 
The derivation of the macroscopic equations will only be sketched here. 
A somewhat more complete discussion will be given in Section 6.10. The 
reason is that for time-varying fields there is a contribution to the atomic 
current from the time derivative of the polarization P. Hence all the 
contributions to the current appear only in the general, time-dependent 
problem. 

The total current density can be divided into : 

(a) conduction-current density J, representing the actual transport of 
charge ; 

(b) atomic-current density J a , representing the circulating currents 
inside atoms or molecules. 

The total vector potential due to all currents is 

a = i f JQO * x ' + i f J .(»'> *f (5.74) 

cJ |x — x'| cJ |x — x'l 

We use a small a for the microscopic vector potential, just as we used e for 
the microscopic electric field in Chapter 4. For the atomic contribution 
we first consider a single molecule, and then average over molecules. The 
discussion proceeds exactly as in Section 5.6 for a localized current 
distribution. For a molecule with center at x, the vector potential at x is 
given approximately by 

a mo i(x) = m ™ix(x-x,) (5 ?5) 

|x - x/ 

To take into account the intrinsic magnetic moments of the electrons, as 
well as the orbital contribution, we interpret m mol as the total molecular 
magnetic moment. If we now sum up over all molecules, averaging as in 
Section 4.3, the macroscopic vector potential can be written 

A(x) _ 1 \J£L rfV + f M <*'> * <' - *'> dV (5.76) 

cJ\x- x'l J |x - x'l 3 

where M(x) is the macroscopic magnetization (magnetic moment per unit 
volume) defined by 

M = AT<m mol > (5*77) 

where N is the number of molecules per unit volume. 



152 Classical Electrodynamics 

The magnetization contribution to A in (5.76) can be rewritten in a more 
useful form : 

JM(x') x (x ~ *'\ d z x' = (m(x') x V d\' (5.78) 

J |x - x'| 3 J |x - x'l 

Then the identity, V x (<f>M) = V0 x M + <f>V x M, can be used to 
obtain 

r M (x<) x &Z^) to _ f^lilM to - IV x (_M_ ) to (5.79) 
J |x-x'| 3 J|x-x'| J \|x-x'|/ 

n x IVI 
The last integral can be converted to a surface integral of , and so 

|x - x'| 

vanishes if M is assumed to be mathematically well behaved and localized 
within a finite volume. Combining the first term in (5.79) with the con- 
duction-current term in (5.76), we can write the vector potential as 

A(x) = lp(x-) + cVxM(x-) d 3,, (580) 

cJ |x — x I 

We see that the magnetization contributes to the vector potential as an 
effective current density J M : 

J M = C (V x M) (5.81) 

There is one questionable step in the derivation of (5.80). That is the 
use of the dipole vector potential (5.75) for all molecules, even those near 
the point x. If a molecule lies within a sphere of radius a few molecular 
diameters d of x, its vector potential will differ appreciably from the dipole 
form (5.75), being much less singular. Thus in (5.80) the contribution 
from that sphere around x is in error. To estimate its importance we note 
that the magnitude of the vector potential per unit volume near x is 
|V x M\/R, while the volume within a distance R to (R + dR) of the 
point x is AttR 2 dR. Hence the contribution to A from the immediate 
neighborhood of x is in error at most by an amount of the order of 
d 2 1 V x M| ~ (d 2 /L) (M), where L is a macroscopic dimension measuring 
the spatial variation of M. Since the whole vector potential is of the order 
of (M)L, the relative error made in using the dipole approximation every- 
where is of the order of d 2 /L 2 . This is completely negligible unless the 
macroscopic length L becomes microscopic ; then the whole development 
fails. 

To obtain the macroscopic equivalent of the curl equation (5.22) we 
calculate B from (5.80) or, what is the same thing, write down (5.22) with 
the total current (J + J M ) replacing J : 

V x B = — J + 4ttV x M (5.82) 

c 



[Sect. 5.8] Magnetostatics 153 

The V x M term can be combined with B to define a new macroscopic 
field H, called the magnetic field, 

H = B - 4ttM (5-83) 

Then the macroscopic equations, replacing (5.26), are 



Att 
V x H = — J 

c 



(5.84) 



V-B = 

The introduction of H as a macroscopic field is completely analogous to 
the introduction of D for the electrostatic field. The macroscopic 
equations (5.84) have their electrostatic counterparts, 

V - D = 4 '" > 1 (5.85) 

V x E = J 

We emphasize that the fundamental fields are E and B. They satisfy the 
homogeneous equations in (5.84) and (5.85). The derived fields, D and H, 
are introduced as a matter of convenience in order to take into account in 
an average way the contributions to p and J of the atomic charges and 
currents. 

In analogy with dielectric media we expect that the properties of magnetic 
media can be described by a small number of constants characteristic of 
the material. Thus in the simplest case we would expect that B and H are 

proportional : 

B = ^H (5.86) 

where fi is a constant characteristic of the material called impermeability.* 
This simple result does hold for materials other than the ferromagnetic 
substances. But for these nonmagnetic materials (x generally differs from 
unity by only a few parts in 10 5 (// > 1 for paramagnetic substances, 
[X < 1 for diamagnetic substances). For the ferromagnetic substances, 
(5.86) must be replaced by a nonlinear functional relationship, 

B = F(H) (5.87) 

The phenomenon of hysteresis, shown schematically in Fig. 5.8, implies 
that B is not a single-valued function of H. In fact, the function F(H) 
depends on the history of preparation of the material. The incremental 
permeability of //(H) is defined as the derivative of B with respect to H, 

* To be consistent with the electrostatic relation D = eE, expressing the derived 
quantity D as a factor times E, we should write H = //B. But traditional usage is that 
of (5.86). It makes most substances have v > 1 . Perhaps that is more comforting than 
H' < 1. 



154 



Classical Electrodynamics 




Fig. 5.8 Hysteresis loop giving B in a 

ferromagnetic material as a function 

ofH. 

assuming that B and H are parallel. For high-permeability substances, 
ju(H) can be as high as 10 6 . Most untreated ferromagnetic materials have 
a linear relation (5.86) between B and H for very small fields. Typical 
values of initial permeability range from 10 to 10 4 . 

The complicated relationship between B and H in ferromagnetic 
materials makes analysis of magnetic boundary- value problems inherently 
more difficult than that of similar electrostatic problems. But the very 
large values of permeability sometimes allow simplifying assumptions on 
the boundary conditions. We will see that explicitly in the next section. 



5.9 Boundary Conditions on B and H 

Before we can solve magnetic boundary-value problems, we must 
establish the boundary conditions satisfied by B and H at the interface 
between two media of different magnetic properties. If a small Gaussian 




Fig. 5.9 



[Sect. 5.9] 



Magnetostatics 



155 



pillbox is oriented so that its faces are in regions 1 and 2 and parallel to 
the surface boundary, S, as shown in Fig. 5.9, Gauss's theorem can be 
applied to V • B = to yield 



(B 2 -B 1 )-n = 



(5.88) 



where n is the unit normal to the surface directed from region 1 into region 
2, and the subscripts refer to values at the surface in the two media. 

If we now consider a small, narrow circuit C, as shown in Fig. 5.9, with 
normal n' parallel to the interface and surface S, Stokes's theorem can be 
applied to the curl equation in (5.84) to give 



<£ H • d\ = — I J • n' 

Jc c Js 



da 



(5.89) 



The contributions to the line integral are the tangential values of H in 
the two regions, while the surface integral is proportional to the surface- 
current density K (charge/length x time) in the limit of vanishing width 
to the loop. Thus (5.89) becomes 



or 



(H 2 - H x ) • (n' x n) = — n' 

c 

n x(H 2 -H!) = — K 



K 



(5.90) 



We express these boundary conditions in terms of the magnetic field H 
and the permeability fi. For simplicity assuming no surface currents, we 
have 



H,.ii=(^)h 1 .ii 
H 2 xn = H 1 x n 



(5.91) 



If (jl x > fx 2 , the normal component of H 2 is much larger than the normal 
component of H l5 as shown in Fig. 5.10. In the limit (/uj^ -** °°> tne 



ll^i:»M2lllllll 



EH 




Fig. 5.10 



156 Classical Electrodynamics 

magnetic field H 2 is normal to the boundary surface, independent of the 
direction of H x (barring the exceptional case of H x exactly parallel to the 
interface). The boundary condition on H at the surface of a very high- 
permeability material is thus the same as for the electric field at the surface 
of a conductor. We may therefore use electrostatic potential theory for 
the magnetic field. The surfaces of the high-permeability material are 
approximately "equipotentials," and the lines of H are normal to these 
equipotentials. This analogy is exploited in many magnet-design problems. 
The type of field is decided upon, and the pole faces are shaped to be 
equipotential surfaces. 



5.10 Uniformly Magnetized Sphere 

To illustrate the different methods possible for the solution of a 
boundary- value problem in magnetostatics, we consider in Fig. 5.11 the 
simple problem of a sphere of radius a, with a uniform permanent 
magnetization M of magnitude M and parallel to the z axis, embedded in 
a nonpermeable medium. Outside the sphere, V • B = V x B = 0. 
Consequently, for r > a, B = H can be written as the negative gradient of 
a magnetic scalar potential which satisfies Laplace's equation, 

B ut= -VO 1 

(5.92) 
V 2 O M = J 

With the boundary condition that B — >■ for r — ► oo, the general solution 
for the potential is ^ 

W) ^ a ,M (5 . 93) 

1 = 

Past experience tells us that only the lowest few terms in this expansion will 
appear, probably just / = 1 . 

Inside a magnetized object we cannot in general use equations (5.92) 
because VxB^O. This causes no difficulty in the present simple situation 
because (5.83) implies that B, H, and M are all parallel in the absence of 
applied fields. 



M = Mo63 




Fig. 5.11 



[Sect. 5.10] 

Hence we assume that 



Magnetostatics 

B in = B o € z 

Hi D = CB - 477M )€3 



157 



(5.94) 



The boundary conditions at the surface of the sphere are that B r and H d 
be continuous. Thus, from (5.92), (5.93), and (5.94), we obtain 



^ (/ + l)a t P,(cos 0) 
B cos =2^ ^Pi 

1 = 

-(fl - 4ttM ) sin = -^ ^Ti ^ 



(5.95) 



Evidently only the / = 1 term survives in the expansion. We find the 
unknown constants ol x and B to be 



4tt w o 
a, = — M a J 
1 3 



5 = ^M 



(5.96) 



The fields outside the sphere are those of a dipole (5.41) of dipole moment, 

(5.97) 



47T o_ , 

m = — afM 

3 



The fields inside are 



B ln = ^M 



Hm= -t^M 



(5.98) 



We note that B in is parallel to M, while H in is antiparallel. The lines of B 
and H are shown in Fig. 5.12. The lines of B are continuous closed paths, 
but those of H terminate on the surface. The surface appears to have a 
"magnetic-charge" density on it. This fictitious charge is related to the 
divergence of the magnetization (see below). 

The solution both inside and outside the sphere could have been 
obtained from electrostatic potential theory if we had chosen to discuss H 
rather than B. We can treat the equations, 



V x H = 
V-H= -477-V-MJ 



(5.99) 



158 



Classical Electrodynamics 





H 



Fig. 5.12 Lines of B and lines of H for a uniformly magnetized sphere. The lines of B 

are closed curves, but the lines of H originate on the surface of the sphere where the 

magnetic "charge," —V • M, resides. 

These equations show that H is derivable from a potential, and that 
— V • M acts as a magnetic-charge density. Thus, with H = — V® M , we 
find 

V 2 ^ = 4ttV • M (5.100) 

Since M is constant in magnitude and direction, its divergence is zero 
inside the sphere. But there is a contribution because M vanishes outside 
the sphere. We write the solution for Q> M inside and outside the sphere as 

<1>m(x) = - f y /,M( ^ } dV (5.101) 

J |x - X | 

Then we use the vector identity V • (<f>M) = M • V<f> + <£V • M to obtain 

<*>m(x) = - f V • -55*1- d*x' + f M(x') • V'(— -L-) dV (5.102) 

J |x — x | J \|x — X |/ 

The first integral vanishes on integration over any volume containing the 
sphere. If we convert the derivative with respect to x' into one with 
respect to x according to the rule V -> — V when operating on any 
function of |x — x'|, the potential can be written 

0> M (x) = - V • f M(X,) d z x' = -V • [m € 3 f V 2 dr' [dQ.' 1 

J |x - x'| L ° 3 J J |x - x'|J 

(5.103) 



[Sect. 5.10] Magnetostatics 159 

Only the / = part of |x — x'l -1 contributes to the integral. Therefore 

O M (x)=-4 7 rM V.[e 3 £^] (5.104) 

The integral yields different values, depending on whether r lies inside or 
outside the sphere. We find easily 

*«(x) = ^f^(^)cose (5.105) 

where r< (r>) is the smaller (larger) of r and a. This potential yields a 
dipole field outside with magnetic moment (5.97) and the constant value 
H in (5.98) inside, in agreement with the first method of solution.* 

Finally we solve the problem using the generally applicable vector 
potential. Referring to (5.80), we see that the vector potential is given by 

A(x) f^xMxO^, (5106) 

J |x — x'| 
Since M is constant inside the sphere, the curl vanishes there. But because 
of the discontinuity of M at the surface, there is a surface integral contri- 
bution to A. If we consider (5.79), the required surface integral can be 
recovered : 

A(x) - - f V x ( M(X,) W = i M(X>) X ° da' (5.107) 
W J \|x - x'|/ J |x - x'l 

The quantity c(M x n) can be considered as a surface-current density. 
The equivalence of a uniform magnetization throughout a certain volume 
to a surface-current density c(M x n) over its surface is a general result 
for arbitrarily shaped volumes. This equivalence is often useful in treating 
fields due to permanent magnets. 

For the sphere with M in the z direction, (M x n) has only an azimuthal 
component, 

(M x n)^ = M sin 0' (5.108) 

To determine A we choose our observation point in the x-z plane for 
calculational convenience, just as in Sections 5.5. Then only the y com- 
ponent of — (n x M) enters. The azimuthal component of the vector 
potential is then 

^(x) = M a4rfQ- S ; ne ' C( */' (5.109) 

J |x — x I 

* The development from (5.101) to (5.105) is unnecessarily complicated for the simple 
calculation at hand. For the uniformly magnetized sphere it is easy to show that 
V- M = -M o cos0<5(r - a). Substitution into (5.101) and use of (3.70) yields 
(5.105) directly. Equation (5.103) is still useful, of course, for more complicated 
distributions of magnetization. 



160 Classical Electrodynamics 

where x' has coordinates (a, 0\ <f>'). The angular factor can be written 

sin 0' cos <j>' = -J— Re[ Y ltl (6', f )] (5.110) 

Thus with expansion (3.70) for |x — x'| only the / = 1, m = 1 term will 
survive. Consequently 

A+(x) = j M a*(^) sind (5.111) 

where r < (r > ) is the smaller (larger) of r and a. With only a <f> component 
of A, the components of the magnetic induction B are given by (5.38). 
Equation (5.111) evidently gives the uniform B inside and the dipole field 
outside, as found before. 

The different techniques used here illustrate the variety of ways of 
solving steady-state magnetic problems, in this case with a specified 
distribution of magnetization. The scalar potential method is applicable 
provided no currents are present. But for the general problem with 
currents we must use the vector potential (apart from special techniques 
for particularly simple geometries). 



5.11 Magnetized Sphere in an External Field; Permanent Magnets 

In Section 5.10 we discussed the fields due to a uniformly magnetized 
sphere. Because of the linearity of the field equations we can superpose a 
uniform magnetic induction 60=110 throughout all space. Then we have 
the problem of a uniformly magnetized sphere in an external field. From 
(5.98) we find that the magnetic induction and field inside the sphere are 
now 



B in = B + — M 
3 

H in = B - jM 



(5.112) 



We now imagine that the sphere is not a permanently magnetized object, 
but rather a paramagnetic or diamagnetic substance of permeability /u. 
Then the magnetization M is a result of the application of the external 
field. To find the magnitude of M we use (5.86): 

B in = /*H in (5.113) 



[Sect. 5.11] 



Magnetostatics 



161 




(5.114) 
(5.115) 



Thus Fig ' 513 

b + |m = / ,(b -^m) 

This gives a magnetization, 

M = M^^)b 
4ttV + 2/ 

We note that this is completely analogous to the polarization P of a 
dielectric sphere in a uniform electric field (4.63). 

For a ferromagnetic substance the arguments of the last paragraph fail. 
Equation (5.115) implies that the magnetization vanishes when the 
external field vanishes. The existence of permanent magnets contradicts 
this result. The nonlinear relation (5.87) and the phenomenon of hysteresis 
allow the creation of permanent magnets. We can solve equations (5.112) 
for one relation between H in and B in by eliminating M: 



B in + 2H in = 3B 



(5.116) 



The hysteresis curve provides the other relation between B in and H in , so 
that specific values can be found for any external field. Equation (5.116) 
corresponds to lines with slope —2 on the hysteresis diagram with inter- 
cepts 3B on the y axis, as in Fig. 5.13. Suppose, for example, that the 
external field is increased until the ferromagnetic sphere becomes saturated 
and decreased to zero. The internal B and H will then be given by the 
point marked P in Fig. 5.13. The magnetization can be found from (5.112) 
with B = 0. 

The relation (5.116) between B in and H in is specific to the sphere. For 
other geometries other relations pertain. The problem of the ellipsoid can 
be solved exactly and shows that the slope of the lines (5.116) range from 
zero for a flat disc to — oo for a long needle-like object. Thus a larger 
internal magnetic induction can be obtained with a rod geometry than 
with spherical or oblate spheroidal shapes. 



1<>2 Classical Electrodynamics 

5.12 Magnetic Shielding; Spherical Shell of Permeable Material in a 
Uniform Field 

Suppose that a certain magnetic induction B exists in a region of empty 
space initially. A permeable body is now placed in the region. The lines 
of magnetic induction are modified. From our remarks at the end of 
Section 5.9 concerning media of very high permeability we would expect 
that the field lines would tend to be normal to the surface of the body. 
Carrying the analogy with conductors further, if the body is hollow, we 
would expect that the field in the cavity would be smaller than the external 
field, vanishing in the limit fj, -► oo. Such a reduction in field is said to be 
due to the magnetic shielding provided by the permeable material. It is of 
considerable practical importance, since essentially field-free regions are 
often necessary or desirable for experimental purposes or for the reliable 
working of electronic devices. 

As an example of the phenomenon of magnetic shielding we consider a 
spherical shell of inner (outer) radius a (b), made of material of perme- 
ability p, and placed in a formerly uniform constant magnetic induction 
B , as shown in Fig. 5.14. We wish to find the fields B and H everywhere 
in space, but most particularly in the cavity (r < a), as functions of p. 
Since there are no currents present, the magnetic field H is derivable from 
a scalar potential, H = -VO M . Furthermore, since B = ^H, the 
divergence equation V . B = becomes V . H = in the various regions. 
Thus the potential <b M satisfies Laplace's equation everywhere. The 
problem reduces to finding the proper solutions in the different regions to 
satisfy the boundary conditions (5.88) and (5.90) at r = a and r = b. 

For r>b, the potential must be of the form, 

oo 

®m = ~B r cos + J -ik ^(cos 0) (5.117) 




Fig. 5.14 



[Sect. 5.12] 



Magnetostatics 



163 



in order to give the uniform field, H = B = B , at large distances. For 
the inner regions, the potential must be 



a < r < 



r < a 



z=o r 



®M = I*ir l P l (cos6) 



1 = 



(5.118) 



The boundary conditions at r — a and r = b are that H e and B r be 
continuous. In terms of the potential O m these conditions become 



9 *-( M =^( M 



do 



dd 



*>* (fl+) _??*(«_) 



do 



dd 



or or 



V 



*>"<«,) = ?*«(«_) 



dr 



dr 



(5.119) 



The notation b ± means the limit r— ►& approached from r ^ b, and 
similarly for a ± . These four conditions, which hold for all angles 6, are 
sufficient to determine the unknown constants in (5.117) and (5.118). All 
coefficients with / ^ 1 vanish. The / = 1 coefficients satisfy the four 
simultaneous equations 



"i- #7*i- 7i =^o 

2a t + fibPp! - 2 f ,iy 1 = -b^B^ 

/^a 3 ^! — 2/^7! — cfid 1 = 0. 
The solutions for a x and 5 X are 

(2/j + !)(// - 1) 



(5.120) 



(2p + 1)0* + 2)- 2^-i/i -l) 2 



(fc 3 - « 3 )5 



<5x=- 



9^ 



(2 /1 + 1)( / i + 2)-2"-( /< -1) 2 



Bn 



(5.121) 



The potential outside the spherical shell corresponds to a uniform field 
B plus a dipole field (5.41) with dipole moment <x x oriented parallel to B . 
Inside the cavity, there is a uniform magnetic field parallel to B and 
equal in magnitude to — d v For /u > 1, the dipole moment a. x and the 



164 



Classical Electrodynamics 




Fig. 5.15 Shielding effect of a shell of highly permeable material. 



inner field — d ± become 



a x — b% 



-*i 



ty 



7 3\ B o 



(5.122) 



We see that the inner field is proportional to fir 1 . Consequently a shield 
made of high-permeability material with fi ~ 10 3 to 10 6 causes a great 
reduction in the field inside it, even with a relatively thin shell. Figure 
5.15 shows the behavior of the lines of B. The lines tend to pass through 
the permeable medium if possible. 



REFERENCES AND SUGGESTED READING 



Problems in steady-state current flow in an extended resistive medium are analogous to 
electrostatic potential problems, with the current density replacing the displacement and 
the conductivity replacing the dielectric constant. But the boundary conditons are 
generally different. Steady-state current flow is treated in 

Jeans, Chapters IX and X, 

Smythe, Chapter VI. 



[Probs. 5] Magnetostatics 165 

Magnetic fields due to specified current distributions and boundary-value problems in 
magnetostatics are discussed, with numerous examples, by 

Durand, Chapters XIV and XV, 

Smythe, Chapters VII and XII. 
The atomic theory of magnetic properties rightly falls in the domain of quantum 
mechanics. Semiclassical discussions are given by 

Abraham and Becker, Band II, Sections 29-34, 

Durand, pp. 551-573, and Chapter XVII, 

Landau and Lifshitz, Electrodynamics of Continuous Media, 

Rosenfeld, Chapter IV. 
Quantum-mechanical treatments appear in books devoted entirely to the electrical and 
magnetic properties of matter, such as 

Van Vleck. 



PROBLEMS 

5.1 Starting with the differential expression 

Id\ X x 

dB = ^~ 

c x A 

for the magnetic induction produced by an increment / d\ of current, show 
explicitly that for a closed loop carrying a current J the magnetic induction 
at an observation point P is 

B = - - VQ 

c 

where O is the solid angle subtended by the loop at the point P. This is an 
alternative form of Ampere's law for current loops. 

5.2 (a) For a solenoid wound with N turns per unit length and carrying a 
current /, show that the magnetic-flux density on the axis is given approxi- 
mately by 

2nNI , „ . 

B z = (cos d 1 + cos 2 ) 



where the angles are defined in the figure. 



(a) 

(b) For a long solenoid of length L and radius a show that near the axis 
and near the center of the solenoid the magnetic induction is mainly parallel 
to the axis, but has a small radial component 

9677JV7/a 2 z/A 

*'~—{-if) 

correct to order a 2 /L 2 , and for z < L, p < a. The coordinate z is measured 
from the center point of the axis. 



166 Classical Electrodynamics 

(c) Show that at the end of a long solenoid the magnetic induction near 
the axis has components 

2-nNI ttNI/p 

B~ ^ , 5„ ^ — 



5.3 A cylindrical conductor of radius a has a hole of radius b bored parallel to, 
and centered a distance </from, the cylinder axis (d + b < a). The current 
density is uniform throughout the remaining metal of the cylinder and is 
parallel to the axis. Use Ampere's law and the principle of linear super- 
position to find the magnitude and the direction of the magnetic-flux 
density in the hole. 

5.4 A circular current loop of radius a carrying a current / lies in the x-y plane 
with its center at the origin. 

(a) Show that the only nonvanishing component of the vector potential is 

Ala f °° 
A^ip, z) = dk cos kz Ijikp^Kjilcp^ 



where p< (p>) is the smaller (larger) of a and p. 
(b) Show that an alternative expression for A+ is 



A+(p, z) = — dk e-*'*l J^Ukp) 



(c) Write down integral expressions for the components of magnetic 
induction, using the expressions of (a) and (b). Evaluate explicitly the 
components of B on the axis by performing the necessary integrations. 
5.5 Two concentric circular loops of radii a, b and currents /, /', respectively 
{b < a), have an angle a between their planes. Show that the torque on one 
of the loops is about the line of intersection of the two planes containing 
the loops and has the magnitude: 



mr 2-^ll'b 2 ^ (» + 1) 

N = 



2 



ac 2 Z-, {In + 1) 

71=0 



T(n + f ) - 
T(n + 2) T(f ) 



2n 



" ^2n+l(COSa) 



where iY(cos a) is an associated Legendre polynomial. Determine the 
sense of the torque for a an acute angle and the currents in the same 
(opposite) directions. 

5.6 A sphere of radius a carries a uniform charge distribution on its surface. 
The sphere is rotated about a diameter with constant angular velocity co. 
Find the vector potential and magnetic-flux density both inside and outside 
the sphere. 

5.7 A long, hollow, right circular cylinder of inner (outer) radius a (b), and of 
relative permeability /j,, is placed in a region of initially uniform magnetic- 
flux density B at right angles to the field. Find the flux density at all points 
in space, and sketch the logarithm of the ratio of the magnitudes of B on the 
cylinder axis to B as a function of log 10 /u for a 2 /b 2 — 0.5, 0. 1 . Neglect end 
effects. 

5.8 A current distribution J(x) exists in a medium of unit permeability adjacent 
to a semi-infinite slab of material having permeability n and filling the 
half-space, z < 0. 



[Probs. 5] Magnetostatics 167 

(a) Show that for z > the magnetic induction can be calculated by 
replacing the medium of permeability ^ by an image current distribution, 
J*, with components, 

(b) Show that for z < the magnetic induction appears to be due to a 
current distribution ( (\ I J in a medium of unit permeability. 

5.9 A circular loop of wire having a radius a and carrying a current / is located 
in vacuum with its center a distance d away from a semi-infinite slab of 
permeability /u. Find the force acting on the loop when 

(a) the plane of the loop is parallel to the face of the slab, 

(b) the plane of the loop is perpendicular to the face of the slab. 

(c) Determine the limiting form of your answers to (a) and (b) when 
d > a. Can you obtain these limiting values in some simple and direct 
way? 

5.10 A magnetically "hard" material is in the shape of a right circular cylinder 
of length L and radius a. The cylinder has a permanent magnetization M , 
uniform throughout its volume and parallel to its axis. 

(a) Determine the magnetic field H and magnetic induction B at all 
points on the axis of the cylinder, both inside and outside. 

O) Plot the ratios B/4ttM and H/4ttM on the axis as functions of z for 
L/a = 5. 

5.11 (a) Starting from the force equation (5.12) and the fact that a magnetiza- 
tion M is equivalent to a current density J M = c(V X M), show that, in 
the absence of macroscopic currents, the total magnetic force on a body 
with magnetization M can be written 



--/■ 



(V • M)B e cPx 



where B e is the magnetic induction due to all other except the one in 
question. 
(b) Show that an alternative expression for the total force is 



--;■ 



(V • M)H d 3 x 



where H is the total magnetic field, including the field of the magnetized 
body. 

Hint: The results of (a) and (b) differ by a self-force term which can be 
omitted (why?). 
5.12 A magnetostatic field is due entirely to a localized distribution of permanent 
magnetization. 

(a) Show that 



B • H d z x = 

provided the integral is taken over all space. 



!■ 



168 Classical Electrodynamics 

ib) From the potential energy (5.73) of a dipole in an external field show 
that for a continuous distribution of permanent magnetization the magneto- 
static energy can be written 



W 



= -!- (h • Ud 3 x = -i MM -Hdh 



apart from an additive constant which is independent of the orientation or 
position of the various constituent magnetized bodies. 

5.13 Show that in general a long, straight bar of uniform cross-sectional area A 
with uniform lengthwise magnetization M, when placed with its flat end 
against an infinitely permeable flat surface, adheres with a force given 
approximately by 

F ~ 2-nAM 2 

5.14 A right circular cylinder of length L and radius a has a uniform lengthwise 
magnetization M. 

(a) Show that, when it is placed with its flat end against an infinitely 
permeable plane surface, it adheres with a force 



F = SiraLM 2 
where 



K(k) - E(k) K(k^) - E(kJ 



K 



* / . „ = > "a — 



V4a 2 + L 2 ' Va 2 + L 2 

(b) Find the limiting form for the force if L :> a. 



6 



Time- Varying Fields, 
Maxwell's Equations, 
Conservation Laws 



In the previous chapters we have dealt with steady-state problems 
in electricity and in magnetism. Similar mathematical techniques were 
employed, but electric and magnetic phenomena were treated as indepen- 
dent. The only link between them was the fact that currents which produce 
magnetic fields are basically electrical in character, being charges in motion. 
The almost independent nature of electric and magnetic phenomena 
disappears when we consider time-dependent problems. Time-varying 
magnetic fields give rise to electric fields and vice-versa. We then must 
speak of electromagnetic fields, rather than electric or magnetic fields. The 
full import of the interconnection between electric and magnetic fields 
and their essential sameness becomes clear only within the framework 
of special relativity (Chapter 1 1). For the present we will content ourselves 
with examining the basic phenomena and deducing the set of equations 
known as Maxwell's equations, which describe the behavior of electro- 
magnetic fields. General properties of these equations will be established 
so that the basic groundwork of electrodynamics will have been laid. 
Subsequent chapters will then explore the many ramifications. 

In our desire to proceed to other things, we will leave out a number of 
topics which, while of interest in themselves, can be studied elsewhere. 
Some of these are quasi-stationary fields, circuit theory, inductance 
calculations, eddy currents, and induction heating. None of these subjects 
involves new concepts beyond what are developed in this chapter and 
previous ones. The interested reader will find references at the end of the 
chapter. 

169 



170 Classical Electrodynamics 

6.1 Faraday's Law of Induction 

The first quantitative observations relating time-dependent electric and 
magnetic fields were made by Faraday (1831) in experiments on the 
behavior of currents in circuits placed in time- varying magnetic fields. It 
was observed by Faraday that a transient current is induced in a circuit 
if (a) the steady current flowing in an adjacent circuit is turned on or off, (b) 
the adjacent circuit with a steady current flowing is moved relative to the 
first circuit, (c) a permanent magnet is thrust into or out of the circuit. No 
current flows unless either the adjacent current changes or there is relative 
motion. Faraday interpreted the transient current flow as being due to a 
changing magnetic flux linked by the circuit. The changing flux induces 
an electric field around the circuit, the line integral of which is called the 
electromotive force, $. The electromotive force causes a current flow, 
according to Ohm's law. 

We now express Faraday's observations in quantitative mathematical 
terms. Let the circuit C be bounded by an open surface S with unit normal 
n, as in Fig. 6.1. The magnetic induction in the neighborhood of the 
circuit is B. The magnetic flux linking the circuit is defined by 



F = B-n da (6.1) 

Js 

The electromotive force around the circuit is 

£=i>E'>dl (6.2) 

where E' is the electric field at the element d\ of the circuit C. Faraday's 
observations are summed up in the mathematical law, 

dF 
£=-k— (6.3) 

dt 

The induced electromotive force around the circuit is proportional to the 
time rate of change of magnetic flux linking the circuit. The sign is 
specified by Lenz's law, which states that the induced current (and 
accompanying magnetic flux) is in such a direction as to oppose the change 
of flux through the circuit. 

The constant of proportionality k depends on the choice of units for the 
electric and magnetic field quantities. It is not, as might at first be 
supposed, an independent empirical constant to be determined from 
experiment. As we will see immediately, once the units and dimensions in 



[Sect. 6.1] Time- Varying Fields, Maxwell's Equations, Conservation Laws 111 




Fig. 6.1 

Ampere's law have been chosen, the magnitude and dimensions of A: follow 
from the assumption of Galilean invariance for Faraday's law. For 
Gaussian units, k = c~ x , where c is the velocity of light. 

Before the development of special relativity (and even afterwards, when 
dealing with relative speeds small compared with the velocity of light), it 
was understood, although not often explicitly stated, by all physicists that 
physical laws should be invariant under Galilean transformations. That 
is, physical phenomena are the same when viewed by two observers 
moving with a constant velocity v relative to one another, provided the 
coordinates in space and time are related by the Galilean transformation, 
x' = x +\t, t' = t. In particular, consider Faraday's observations. It is 
obvious (i.e., experimentally verified) that the same current is induced in a 
circuit whether it is moved while the circuit through which current is 
flowing is stationary or it is held fixed while the current-carrying circuit is 
moved in the same relative manner. 

Let us now consider Faraday's law for a moving circuit and see the 
consequences of Galilean invariance. Expressing (6.3) in terms of the 
integrals over E' and B, we have 

<pE' -dl= -k— B-nda (6.4) 

Jc dt Js 

The induced electromotive force is proportional to the total time derivative 

of the flux — the flux can be changed by changing the magnetic induction 

or by changing the shape or orientation or position of the circuit. In form 

(6.4) we have a far-reaching generalization of Faraday's law. The circuit 

C can be thought of as any closed geometrical path in space, not necessarily 

coincident with an electric circuit. Then (6.4) becomes a relation between 

the fields themselves. It is important to note, however, that the electric 

field, E' is the electric field at d\ in the coordinate system in which d\ is at 

rest, since it is that field which causes current to flow if a circuit is actually 

present. 



172 Classical Electrodynamics 

If the circuit C is moving with a velocity v in some direction, as shown in 
Fig. 6.2, the total time derivative in (6.4) must take into account this 
motion. The flux through the circuit may change because (a) the flux 
changes with time at a point, or (b) the translation of the circuit changes 
the location of the boundary. It is easy to show that the result for the total 
time derivative of flux through the moving circuit is* 

— \B-nda = \^-nda + <k(Bxv)'dl (6.5) 

dt Js Js dt Jc 

Equation (6.4) can now be written in the form, 

(f> [E' - fe(v x B)] • d\ = -k f — • n da (6.6) 

Jc Js dt 

This is an equivalent statement of Faraday's law applied to the moving 
circuit C. But we can choose to interpret it differently. We can think of 
the circuit C and surface S as instantaneously at a certain position in space 
in the laboratory. Applying Faraday's law (6.4) to that fixed circuit, we 
find 

i>E-dl= -k\ —-nda (6.7) 

Jc Js dt 

where E is now the electric field in the laboratory. The assumption of 
Galilean invariance implies that the left-hand sides of (6.6) and (6.7) must 
be equal. This means that the electric field E' in the moving coordinate 
system of the circuit is 

E' = E + k(y x B) (6.8) 

To determine the constant k we merely observe the significance of E\ A 
charged particle (e.g., one of the conduction electrons) in a moving circuit 




Fig. 6.2 



1 



For a general vector field there is an added term, I (V • B)v • n da, which gives the 

Js 

contribution of the sources of the vector field swept over by the moving circuit. The 
general result follows most easily from the use of the convective derivative, 

d d 

-T=-+VV 
dt dt 



[Sect. 6.2] Time-Varying Fields, Maxwell's Equations, Conservation Laws 173 

will experience a force qE'. When viewed from the laboratory, the charge 
represents a current J = qs d(x — x ). From the magnetic force law (5.7) 
or (5.12) it is evident that this current experiences a force in agreement 
with (6.8) provided the constant k is equal to c _1 . 
We have thus reached the conclusion that Faraday's law takes the form 

(f> E'dl = --- \*>nda (6.9) 

Jc c dt Js 

where E' is the electric field at d\ in its rest frame of coordinates. The time 
derivative on the right is a total time derivative. If the circuit C is moving 
with a velocity v, the electric field in the moving frame is 

E' = E + - (v x B) (6.10) 

c 

These considerations are valid only for nonrelativistic velocities. Galilean 
invariance is not rigorously valid, but holds only for relative velocities 
small compared to the velocity of light. Expression (6.10) is correct to 
first order in vjc, but in error by terms of order v 2 /c 2 (see Section 11.10). 
Evidently, for laboratory experiments with macroscopic circuits, (6.9) and 
(6.10) are completely adequate. 

Faraday's law (6.9) can be put in differential form by use of Stokes's 
theorem, provided the circuit is held fixed in the chosen reference frame 
(in order to have E and B defined in the same frame). The transformation 
of the electromotive force integral into a surface integral leads to 



/,( 



VxE + - — )-nda=0 

s\ c dt 



Since the circuit C and bounding surface S are arbitrary, the integrand 
must vanish at all points in space. 
Thus the differential form of Faraday's law is 

VXE + -- = (6.11) 

c dt 

We note that this is the time-dependent generalization of the statement, 
V x E = 0, for electrostatic fields. 



6.2 Energy in the Magnetic Field 

In discussing steady-state magnetic fields in Chapter 5 we avoided the 
question of field energy and energy density. The reason was that the 
creation of a steady-state configuration of currents and associated magnetic 



174 Classical Electrodynamics 

fields involves an initial transient period during which the currents and 
fields are brought from zero to their final values. For such time-varying 
fields there are induced electromotive forces which cause the sources of 
current to do work. Since the energy in the field is by definition the total 
work done to establish it, we must consider these contributions. 

Suppose for a moment that we have only a single circuit with a constant 
current / flowing in it. If the flux through the circuit changes, an electro- 
motive force $ is induced around it. In order to keep the current constant, 
the sources of current must do work at the rate, 

dt c dt 

This is in addition to ohmic losses in the circuit which are not to be 
included in the magnetic-energy content. Thus, if the flux change through 
a circuit carrying a current / is 6F, the work done by the sources is 

6W=-I6F 
c 

Now we consider the problem of the work done in establishing a general 
steady-state distribution of currents and fields. We can imagine that the 
build-up process occurs at an infinitesimal rate so that V • J = holds to 
any desired degree of accuracy. Then the current distribution can be 
broken up into a network of elementary current loops, the typical one of 
which is an elemental tube of current of cross-sectional area Ac following 
a closed path C and spanned by a surface 5" with normal n, as shown in 
Fig. 6.3. 

We can express the increment of work done against the induced emf in 
terms of the change in magnetic induction through the loop : 

A(dW) = t^-\n-dBda 
c Js 

where the extra A comes from the fact that we are considering only one 
elemental circuit. If we express B in terms of the vector potential A, 
then we have 



c Js 



A(dW) = -=Z (V x <5A) • n da 
c Js 

With application of Stokes's theorem this can be written 
A(W) = J-^-&>SK-d\ 

C Jn 



[Sect. 6.2] Time- Varying Fields, Maxwell's Equations, Conservation Laws 175 




Fig. 6.3 Distribution of current 

density broken up into elemental 

current loops. 



But J Act d\ is equal to J d 3 x, by definition, since d\ is parallel to J. 
Evidently the sum over all such elemental loops will be the volume 
integral. Hence the total increment of work done by the external sources 
due to a change 6A(x) in the vector potential is 



6W 



- l .S 



dA • J d 3 x 



(6.12) 



An expression involving the magnetic fields rather than J and dA can be 
obtained by using Ampere's law : 



Att 
V x H= — J 



Then 



dW=— |<5A.(V x U)d 3 x 
Att J 

The vector identity, 

V • (P x Q) = Q • (V x P) - P • (V x Q) 

can be used to transform (6.13): 



(6.13) 



dW 



= J_ f[H . (V x <5A) + V • (H x dA)] d 3 x (6.14) 

Att J 



If the field distribution is assumed to be localized, the second integral 
vanishes. With the definition of B in terms of A, the energy increment can 
be written : 



6W 



= -L|h 

Att J 



• 6B d 3 x 



(6.15) 



This relation is the magnetic equivalent of the electrostatic equation (4.89). 
In its present form it is applicable to all magnetic media, including ferro- 
magnetic substances. If we assume that the medium is para- or dia- 
magnetic, so that a linear relation exists between H and B, then 

H • 6B = £ (5(H . B) 



176 Classical Electrodynamics 

If we now bring the fields up from zero to their final values, the total 
magnetic energy will be 

W=— \n-Bd 3 x (6.16) 

This is the magnetic analog of (4.92). 

The magnetic equivalent of (4.86) where the electrostatic energy is 
expressed in terms of charge density and potential, can be obtained from 
(6.12) by assuming a linear relation between J and A. Then we find the 
magnetic energy to be 

W=—]J'Ad z x (6.17) 

2c J 

The magnetic problem of the change in energy when an object of 
permeability fi x is placed in a magnetic field whose current sources are 
fixed can be treated in close analogy with the electrostatic problem of 
Section 4.8. The role of E is taken by B, that of D by H. The original 
medium has permeability ju and existing magnetic induction B . After 
the object is in place the fields are B and H. It is left as an exercise for the 
reader to verify that for fixed sources of the field the change in energy is 

1^=— (BH -HB )d 3 :r (6.18) 

S7TJVi 

where the integration is over the volume of the object. This can be written 
in the alternative forms : 

W= i- f (^ - ^H • Ho d*x = i- f (1 - i) B • B d*x (6.19) 

Both fi x and /u can be functions of position, but they are assumed inde- 
pendent of field strength. 

If the object is in otherwise free space (/u = 1), the change in energy can 
be expressed in terms of the magnetization as 

W = I M • B d 3 x (6.20) 

JVx 

It should be noted that (6.20) is equivalent to the electrostatic result 
(4.96), except for sign. This sign change arises because the energy W 
consists of the total energy change occurring when the permeable body is 
introduced in the field, including the work done by the sources against the 
induced electromotive forces. In this respect the magnetic problem with 
fixed currents is analogous to the electrostatic problem with fixed potentials 
on the surfaces which determine the fields. By an analysis equivalent to 



[Sect. 6.3] Time-Varying Fields, Maxwell's Equations, Conservation Laws 177 

that at the end of Section 4.8 we can show that for a small displacement 
the work done against the induced emf 's is twice as large as, and of the 
opposite sign to, the potential-energy change of the body. Thus, to find 
the force acting on the body, we consider a generalized displacement £ and 
calculate the positive derivative of W with respect to the displacement: 

'.-©, 

The subscript J implies fixed source currents. 

The difference between (6.20) and the potential energy (5.73) for a 
permanent magnetic moment in an external field (apart from the factor \, 
which is traced to the linear relation assumed between M and B) comes 
from the fact that (6.20) is the total energy required to produce the con- 
figuration, whereas (5.73) includes only the work done in establishing the 
permanent magnetic moment in the field, not the work done in creating 
the magnetic moment and keeping it permanent. 



6.3 Maxwell's Displacement Current; Maxwell's Equations 

The basic laws of electricity and magnetism which we have discussed so 
far can be summarized in differential form by these four equations : 

Coulomb's law: V • D = A-np 

Att 
Ampere's law: V x H = — J 



c 



- (6.22) 



Faraday's law: V x E + - — = 

c dt 

Absence of free magnetic poles : V • B = 

These equations are written in macroscopic form and in Gaussian units. 
Let us recall that all but Faraday's law were derived from steady-state 
observations. Consequently, from a logical point of view there is no a 
priori reason to expect that the static equations hold unchanged for time- 
dependent fields. In fact, the equations in set (6.22) are inconsistent as 
they stand. 

It required the genius of J. C. Maxwell, spurred on by Faraday's 
observations, to see the inconsistency in equations (6.22) and to modify 
them into a consistent set which implied new physical phenomena, at that 
time unknown but subsequently verified in all details by experiment. For 
this brilliant stroke in 1865, the modified set of equations is justly known as 
Maxwell's equations. 



178 Classical Electrodynamics 

The faulty equation is Ampere's law. It was derived for steady-state 
current phenomena with V • J = 0. This requirement on the divergence 
of J is contained right in Ampere's law, as can be seen by taking the diver- 
gence of both sides : 

-V.J = V.(VxH)sO (6.23) 

While V • J = is valid for steady-state problems, the complete relation 
is given by the continuity equation for charge and current : 

V • J + f£ = (6.24) 

ot 

What Maxwell saw was that the continuity equation could be converted 
into a vanishing divergence by using Coulomb's law (6.22). Thus 

Then Maxwell replaced J in Ampere's law by its generalization, 

J ^ J + ^¥ (6 - 26 > 

for time-dependent fields. Thus Ampere's law became 

VXH = ^ J + If (6 . 27) 

c c ot 

still the same, experimentally verified, law for steady-state phenomena, 
but now mathematically consistent with the continuity equation (6.24) for 
time-dependent fields. Maxwell called the added term in (6.26) the 
displacement current. This necessary addition to Ampere's law is of crucial 
importance for rapidly fluctuating fields. Without it there would be no 
electromagnetic radiation, and the greatest part of the remainder of this 
book would have to be omitted. It was Maxwell's prediction that light 
was an electromagnetic wave phenomenon, and that electromagnetic 
waves of all frequencies could be produced, which drew the attention of all 
physicists and stimulated so much theoretical and experimental research 
into electromagnetism during the last part of the nineteenth century. 
The set of four equations, 



c c dt 

V • B = VxE + -— = 

c dt 



(6.28) 



[Sect. 6.4] Time-Varying Fields, Maxwell's Equations, Conservation Laws 179 

known as Maxwell's equations, forms the basis of all electromagnetic 
phenomena. When combined with the Lorentz force equation and 
Newton's second law of motion, these equations provide a complete 
description of the classical dynamics of interacting charged particles and 
electromagnetic fields (see Section 6.9 and Chapters 10 and 12). For 
macroscopic media the dynamical response of the aggregates of atoms is 
summarized in the constitutive relations which connect D and J with E, 
and H with B (e.g., D = eE, J = <rE, B = /M for an isotropic, permeable, 
conducting dielectric). 

The units employed in writing Maxwell's equations (6.28) are those of 
the previous chapters, namely, Gaussian. For the reader more at home in 
other units, such as mks, Table 2 of the Appendix summarizes essential 
equations in the commoner systems. Table 3 of the Appendix allows the 
conversion of any equation from Gaussian to mks units, while Table 4 
gives the corresponding conversions for given amounts of any variable. 



6.4 Vector and Scalar Potentials 

Maxwell's equations consist of a set of coupled first-order partial 
differential equations relating the various components of electric and 
magnetic fields. They can be solved as they stand in simple situations. 
But it is often convenient to introduce potentials, obtaining a smaller 
number of second-order equations, while satisfying some of Maxwell's 
equations identically. We are already familiar with this concept in 
electrostatics and magnetostatics, where we used the scalar potential <E> and 
the vector potential A. 

Since V • B = still holds, we can define B in terms of a vector potential : 

B = V x A (6.29) 

Then the other homogeneous equation in (6.28), Faraday's law, can be 
written 

V x (e + -— ) =0 (6.30) 

\ c dt! 

This means that the quantity with vanishing curl in (6.30) can be written 
as the gradient of some scalar function, namely, a scalar potential O : 



c dt 

c dt 



(6.31) 



180 Classical Electrodynamics 

The definition of B and E in terms of the potentials A and <t> according to 
(6.29) and (6.31) satisfies identically the two homogeneous Maxwell's 
equations. The dynamic behavior of A and O will be determined by the 
two inhomogeneous equations in (6.28). 

At this stage it is convenient to restrict our considerations to the 
microscopic form of Maxwell's equations. Then the inhomogeneous 
equations in (6.28) can be written in terms of the potentials as 

V 2 0> + - |- (V • A) = -4tt P (6.32) 

cot 

We have now reduced the set of four Maxwell's equations to two equations. 
But they are still coupled equations. The uncoupling can be accomplished 
by exploiting the arbitrariness involved in the definition of the potentials. 
Since B is defined through (6.29) in terms of A, the vector potential is 
arbitrary to the extent that the gradient of some scalar function A can be 
added. Thus B is left unchanged by the transformation, 

A -* A' = A + VA (6.34) 

In order that the electric field (6.31) be unchanged as well, the scalar 
potential must be simultaneously transformed, 

<D-*<D' = <D_I^ (6.35) 

c dt 

The freedom implied by (6.34) and (6.35) means that we can choose a set 
of potentials (A, <D) such that 

1 dO 
V-A + -— = (6.36) 

c ot 

This will uncouple the pair of equations (6.32) and (6.33) and leave two 
inhomogeneous wave equations, one for O and one for A: 



™-Y-^-=-4"P (6-37) 



Equations (6.37) and (6.38), plus (6.36), form a set of equations equivalent 
in all respects to Maxwell's equations. 



[Sect. 6.5] Time-Varying Fields, Maxwell's Equations, Conservation Laws 181 

6.5 Gauge Transformations; Lorentz Gauge; Coulomb Gauge 

The transformation (6.34) and (6.35) is called a gauge transformation, 
and the in variance of the fields under such transformations is called gauge 
invar iance. The relation (6.36) between A and <I> is called the Lorentz 
condition. To see that potentials can always be found to satisfy the 
Lorentz condition, suppose that the potentials A, O which satisfy (6.32) 
and (6.33) do not satisfy (6.36). Then let us make a gauge transformation 
to potentials A', O' and demand that A', <!>' satisfy the Lorentz condition: 

V.A' + i^-O-V.A + i^ + VA-i^- (6.39) 

c dt c dt c 2 dr 

Thus, provided a gauge function A can be found to satisfy 

the new potentials A', O' will satisfy the Lorentz condition and the wave 
equations (6.37) and (6.38). 

Even for potentials which satisfy the Lorentz condition (6.36) there is 
arbitrariness. Evidently the restricted gauge transformation, 



(6.41) 



where V 2 A-^^ = (6.42) 

c 2 dr 

preserves the Lorentz condition, provided A, $ satisfy it initially. All 
potentials in this restricted class are said to belong to the Lorentz gauge. 
The Lorentz gauge is commonly used, first because it leads to the wave 
equations (6.37) and (6.38) which treat <D and A on equivalent footings, 
and second because it is a concept which is independent of the coordinate 
system chosen and so fits naturally into the considerations of special 
relativity (see Section 11.9). 

Another useful gauge for the potentials is the so-called Coulomb or 
transverse gauge. This is the gauge in which 

V • A = (6.43) 



A^A + VA 




^ ^ 13A 
c dt 


" 


c*dt* 






182 Classical Electrodynamics 

From (6.32) we see that the scalar potential satisfies Poisson's equation, 

V 2 0> = -4tt/) (6.44) 

with solution, 

0(x, i) = [ p(x '' ° d 3 x' (6.45) 

J |x — x'| 

The scalar potential is just the instantaneous Coulomb potential due to 
the charge density />(x, t). This is the origin of the name "Coulomb 
gauge." 

The vector potential satisfies the inhomogeneous wave equation, 



c 2 dt 2 c c dt 



V 2 A-i-|f=-^J + ±V^ (6.46) 



The "current" term involving the potential can, in principle, be calculated 
from (6.45). Formally, we use the continuity equation to write 

v ^ = _ v r^j(x^)^ {6A7) 

dt J |x-x'| v } 

If the current is written as the sum of a longitudinal and transverse part, 

J = J* + J, (6.48) 

where V x J, = and V • J t = 0, then the parts can be written 

J < = -f v fr-^i dV (6 - 49 > 

477 J |x — x I 

J f =7 VxVx l^-T i d * x ' ( 6 - 50 > 

47T J |x — x I 

This can be proved by using the vector identity, V x (V x J) = 
V(V • J) - V 2 J, together with V 2 (l/|x - x'|) = -4tt <5(x - x'). Com- 
parison of (6.47) with (6.49) shows that 

V^ = 4ttJ 1 (6.51) 

at 

Therefore the source for the wave equation for A can be expressed entirely 
in terms of the transverse current (6.50): 

V2A -V^=--J. (6-52) 

- <T Ot ■ C 

This is, of course, the origin of the name "transverse gauge." 



[Sect. 6.6] Time-Varying Fields, Maxwell's Equations, Conservation Laws 183 

The Coulomb or transverse gauge is often used when no sources are 

present. Then 0> = 0, and A satisfies the homogeneous wave equation. 

The fields are given by 

13A1 
E= -- — 

c dt 



B = V x A 



(6.53) 



6.6 Green's Function for the Time-Dependent Wave Equation 

The wave equations (6.37), (6.38), and (6.52) all have the basic structure, 

V>_iff=-4 7 r/(x,0 (6-54) 

r c 2 dt 2 

where /(x, is a known source distribution. The factor c is the velocity 
of propagation in the medium. 

To solve (6.54) it is useful to find a Green's function for the equation, 
just as in electrostatics. Since the time is involved, the Green's function 
will depend on the variables (x, x\ t, t'), and will satisfy the equation, 

( Vx 2 - i ^G(x, t ; x', t') = -4tt d(x - x') b{t - t') (6.55) 

Then in infinite space with no boundary surfaces the solution of (6.54) 
will be 

V<x, = Jg(x, t ; x', t')f(x', t') d*x' dt' (6.56) 

Of course, the Green's function will have to satisfy certain boundary 
conditions demanded by physical considerations. 

The basic Green's function satisfying (6.55) is a function only of the 
differences in coordinates (x — x') and times (t - t'). To find G we 
consider the Fourier transform of both sides of (6.55). The delta functions 
on the right have the representation, 

<5(x - x') S(t - O = -i- 4 |>fc fW k( ~V-<-> (6.57) 
(2tt) J J 

We therefore write the representation of G as 

G(x, V, x', O = ja*k J^g(k, a>y k -<~ V^-<'> (6.58) 



184 Classical Electrodynamics 

The Fourier transform g(k, co) is to be determined. When (6 57) and 
(6.58) are substituted into the defining equation (6.55), it turns out that 
g(k, co) is 

1 1 



g(k, co) = 



4tt 3 



(* 2 -?) 



(6.59) 



When g(k, co) is substituted into (6.58) and the integrations over k and 
co are begun, there appears a singularity in the integrand at k 2 = co 2 /c 2 
Consequently solution (6.59) is meaningless without some rule as to how 
to handle the singularities. The rule cannot come from the mathematics. 
It must come from physical considerations. The Green's function satis- 
fying (6.55) represents the wave disturbance caused by a point source at x' 
which is turned on only for an infinitesimal time interval at t' = t. We 
know that such a wave disturbance propagates outwards as a spherically 
diverging wave with a velocity c. Hence we demand that our solution for 
G have the following properties : 

(a) G = everywhere for t < t'. 

(b) G represent outgoing waves for t > t' . 

If we think of the co integration in (6.58), the singularities in^(k, co) occur 
at co = ±ck. We can do the co integration as a Cauchy integral in the 
complex co plane. For t > t' the integral along the real axis in (6.58) is 
equivalent to the contour integral around a path C closed in the lower 
half-plane, since the contribution on the semicircle at infinity vanishes 
exponentially. On the other hand, for t' > t, the contour must be closed 
in the upper half-plane, as shown in Fig. 6.4 by path C. 

In order to make G vanish for t < t' we must imagine that the poles at 
co = ±ck are displaced below the real axis, as in Fig. 6.4. Then the integral 
over C for t > t' will give a nonvanishing contribution, while the integral 



t<t' 




Fig. 6.4 Complex to plane with contour 
C for / > /' and contour C for t < t'. 



[Sect. 6.6] Time-Varying Fields, Maxwell's Equations, Conservation Laws 185 

over C for t < t' will vanish. The displacement of the poles can be 
accomplished mathematically by writing (w + ie) in place of a) in (6.59). 
Then the Green's function is given by 

/. /• ik-R- ia>T 

G(x, t; x\ t') = — I d*k\ da> e - (6 M) 

c 2 

where R = x — x', t = t — t', and e is a positive infinitesimal. 

The integration over co for t > can be done with Cauchy's theorem 
applied around the contour C of Fig. 6.4, giving 

G = JL(a* k j**™LlciV (6.61) 

2tt*J k 

The integration over d z k can be accomplished by first integrating over 
angles. Then 

G = — \dk sin (kR) sin (cr/c) (6.62) 

Since the integrand is even in k, the integral can be written over the whole 
interval, — oo < k < oo. With a change of variable x = ck, (6.62) can 
be written 

G = -J— f °° <te(e i[r " (B/c)]a! — e i[r+(B/c)]a! ) (6.63) 

From (2.52) we see that the integrals are just Dirac delta functions. The 
argument of the second one never vanishes (remember, t > 0). Hence 
only the first integral contributes, and the Green's function is 

or, more explicitly, 

G(x, v, x', O = -^— : -, " < 6 - 64 > 

|x - x'| 

This Green's function is sometimes called the retarded Green's function 
because it exhibits the causal behavior associated with a wave disturbance. 
The effect observed at the point x at time t is due to a disturbance which 

|x - x'| . . . 
originated at an earlier or retarded time t' = t at the point x . 



186 Classical Electrodynamics 

The solution for the wave equation (6.54) in the absence of boundaries 



is 



ip(x, t) = 



y c -Lfv.ndv* (6 - 65) 



X — X 



The integration over dt' can be performed to yield the so-called "retarded 
solution," 

^(x,0 = f^^% t dV (6.66) 

J |x — x | 

The square bracket [ ] ret means that the time t' is to be evaluated at the 

retarded time, t' = t — * . 



6.7 Initial-Value Problem; Kirchhoff's Surface-Integral Representation 

Solution (6.66) is a particular integral of the inhomogeneous wave 
equation (6.54). To it can be added any solution of the homogeneous 
wave equation necessary to satisfy the boundary conditions. From the 
table at the end of Section 1.9 we see that the proper boundary conditions 
are Cauchy boundary conditions (ip and dy/dn given) on an "open surface." 
For the three-dimensional wave equation an open surface is defined as a 
three-dimensional volume specified by one functional relationship between 
the four coordinates (x, y, z, t). The customary open surface is ordinary 
three-dimensional space at a fixed time, t = t . Then the problem is an 
initial-value problem with y>(x, t ) and dy(x, t )/dt given for all x. We 
wish to determine ip(x, i) for all times t > t . 

To discuss the initial-value problem and also an integral representation 
of Kirchhoff for closed bounding surfaces, we use Green's theorem (1.35), 
integrated over time from t' = t to t' = t x > t: 

[ V [ <*V(0 V V - V V V) = f V iL%L- v *±) da' (6.67) 

We now choose yj = y> and <f> = G. With wave equations (6.54) and (6.55) 
the left-hand side can be written 

L-HS =J l dt'j dVUrrvKx', t') d(x' - x) d(t' - t) 

- W .OC + i(eji-rg)] (6.68) 



[Sect. 6.7] Time-Varying Fields, Maxwell's Equations, Conservation Laws 187 

The first two terms in (6.68) will evidently give the particular integral (6.66). 
The last two terms can be integrated by parts with respect to the time to 
give 

L-HS = 4TTtp(x, t) - 4tt j V j dV/(x\ t')G 

c 2 Jr \ dt' dt r=t 

Since G = at t' = t x > t, the upper limit vanishes. We can thus combine 
(6.69) and (6.67) to give the integral representation for y(x, i) inside the 
volume V, bounded by the surface S, at times t> t : 

We have written the first term in (6.70) in the usual form (6.66) by using 
the explicit result (6.64) for G. We now do the same for the other terms. 
For simplicity, first we consider the infinite-domain initial-value 
problem with xp and dxpfdt as given functions of space at t = t = 0: 



V,(x,0) = F(x), ^(x,0)=D(x) 
at 



(6.71) 



Then the surface integral in (6.70) can be omitted. To simplify the 
notation we take the observation point at the origin and use spherical 
coordinates in the integrals. Then we have 



[D(/ >Q o4--')- F(r '> Q) l4' + 7- r )L 



X 

The derivative of the delta function can be written 

= c 2 d'(r' - ct) 



(6.72) 



M-H 



t'=0 



Then, with the properties of the delta function summarized in Section 1.2, 
(6.72) becomes 



Y<0, = fdQ' J*°V dr'f[r', a',t' = t-^J 

+ -L La[tD(ct, a r ) + 1 (tF(ct, Q'» 



(6.73) 



188 Classical Electrodynamics 

This is called PoissorCs solution to the initial-value problem. With no 
sources present (/= 0), only values of the initial field at distances ct from 
the origin contribute at time t. 

The initial-value problem for the wave equation has been extensively 
studied in one, two, three, and more dimensions. The reader is referred to 
Morse and Feshbach, pp. 843-847, and to the more mathematical treat- 
ment of Hadamard. 

The other result which we wish to obtain from (6.70) is the so-called 
Kirchhoff representation of the field inside the volume V in terms of the 
values of %p and its derivatives on the boundary surface S. We thus assume 
that there are no sources within V and that the initial values of y and 
dxpjdt vanish. (Alternatively, we can assume that the initial time is in the 
remote past so that there are no more contributions from the initial-value 
solution (6.73) within the volume V.) Then the field inside V is given by 

y>(x, = J- f V £> da'io ^- - y, *2) (6.74) 

47rJt Js \ dn' Y dn'J K J 

With G given by (6.64) we can calculate dG/dn' : 



dR RdR 



R 



.( ^H , 4-+H ) 

*l R 2 cR I 



(6.75) 



The term involving the derivative of the delta function can be integrated 
by parts with respect to the time t' . Then the Kirchhoff integral repre- 
sentation is 

AttJs I R R 3 * y J C R % dt' Jret 

(6.76) 

where R = x — x', and n is a unit normal to the surface S. We emphasize 
that (6.76) is not a solution for the field xp, but only an integral representa- 
tion in terms of its value and the values of its space and time derivatives on 
the surface. These cannot be specified arbitrarily; they are known only 
when the appropriate Cauchy boundary-value problem has been solved. 

The Kirchhoff integral (6.76) is a mathematical statement of Huygens' 
principle and is used as the starting point in discussing optical-diffraction 
problems. Diffraction is discussed in detail in Chapter 9, Section 9.5 and 
below. 



[Sect. 6.8] Time-Varying Fields, Maxwell's Equations, Conservation Laws 189 

6.8 Poynting's Theorem 

The forms of the laws of conservation of energy and momentum are 
important results to establish for the electromagnetic field. We begin by 
considering conservation of energy, often called Poynting's theorem (1884). 
For a single charge q the rate of doing work by external electromagnetic 
fields E and B is qv • E, where v is the velocity of the charge. The magnetic 
field does no work, since the magnetic force is perpendicular to the velocity. 
If there exists a continuous distribution of charge and current, the total 
rate of doing work by the fields in a finite volume V is 

f J.Ed 3 * (6.77) 

Jv 

This power represents a conversion of electromagnetic energy into 
mechanical or thermal energy. It must be balanced by a corresponding 
rate of decrease of energy in the electromagnetic field within the volume V. 
In order to exhibit this conservation law explicitly, we will use Maxwell's 
equations to express (6.77) in other terms. Thus we use the Ampere- 
Maxwell law to eliminate J: 

f J.Ed 3 x = -f [cE.(VxH)-E.^l^ (6.78) 
J v AttJv L ot J 

If we now employ the vector identity, 

V • (E x H) = H • (V x E) - E • (V x H) 

and use Faraday's law, the right side of (6.78) becomes 

l j - E ^=^H cv - (ExH)+E -f +H -f 



d 3 x (6.79) 



To proceed further we must make two assumptions. The first one is not 
fundamental, and is made for simplicity only. We assume that the macro- 
scopic medium involved is linear in its electric and magnetic properties. 
Then the two time derivatives in (6.79) can be interpreted, according to 
equations (4.92) and (6.16), as the time derivatives of the electrostatic and 
magnetic energy densities. We now make our second assumption, namely, 
that the sum of (4.92) and (6.16) represents the total electromagnetic 
energy, even for time-varying fields. Then if the total energy density is 
denoted by 

u = — (E • D + B • H) (6.80) 



190 Classical Electrodynamics 

(6.79) can be written 



-f J.Ed 3 *=f [^ + -lv.(ExH) 

JV JV Lot 477 



d 3 x (6.81) 



Since the volume V is arbitrary, this can be cast into the form of a dif- 
ferential continuity equation or conservation law, 

^ + V-S=-J.E (6.82) 

The vector S, representing energy flow, is called Poynting's vector. It is 
given by 

S = .JL ( E x H) (6.83) 

47T 

and has the dimensions of (energy/area x time). Since only its divergence 
appears in the conservation law, Poynting's vector is arbitrary to the 
extent that the curl of any vector field can be added to it. Such an added 
term can, however, have no physical consequences. Hence it is customary 
to make the specific choice (6.83). 

The physical meaning of the integral or differential form (6.81) or (6.82) 
is that the time rate of change of electromagnetic energy within a certain 
volume, plus the energy flowing out through the boundary surfaces of the 
volume per unit time, is equal to the negative of the total work done by the 
fields on the sources within the volume. This is the statement of conser- 
vation of energy. If nonlinear effects, such as hysteresis in ferromagnetic 
materials, are envisioned, the simple law (6.82) is no longer valid, but 
must be supplemented by terms giving the hysteresis power loss. 



6.9 Conservation Laws for a System of Charged Particles 
and Electromagnetic Fields 

The statements (6.81) and (6.82) of Poynting's theorem have empha- 
sized the energy of the electromagnetic fields. The work done per unit time 
per unit volume by the fields (J • E) is a conversion of electromagnetic into 
mechanical or heat energy. Since matter is ultimately composed of 
charged particles (electrons and atomic nuclei), we can think of this rate 
of conversion as a rate of increase of energy of the charged particles per unit 
volume. Then we can interpret Poynting's theorem for the microscopic 
fields as a statement of conservation of energy of the combined system of 
particles and fields. If we denote the total energy of the particles within 



[Sect. 6.9] Time- Varying Fields, Maxwell's Equations, Conservation Laws 191 

the volume V as E mech and assume that no particles move out of the 

volume, we have ,. 

™2™* = \ J.EcPx (6.84) 

dt Jv 

Then Poynting's theorem expresses the conservation of energy for the 
combined system as 

^ = 4 ( £ mech + Efleld) = - <P n • S da (6.85) 

dt dt Js 

where the total field energy within V is 

£ fl eid = I u d z x = — I (E 2 + B 2 ) d z x (6.86) 

Jv SttJv 

The conservation of linear momentum can be similarly considered. We 
have seen that the force on a charge q in an external field E is qE. From the 
basic law (5.12) for forces on currents we can deduce that the magnetic 
force on a charge q moving with velocity v in an external magnetic induction 
B is (q/c) v x B. Thus the total electromagnetic force on a charged particle 

18 / v \ 

F=tf(E + -xBl (6.87) 

This is called the Lor entz force. Although we have deduced it within the 
framework of steady-state phenomena, it is well verified for all charged 
particles with arbitrarily large velocities. 

From Newton's second law we can write the rate of change of the 
particle's momentum as 

jr q ( E+ l* B ) (6 - 88) 

If the sum of all the momenta of all the particles in the volume V is 
denoted by P mech , we can write 



= ( P E +ij x B)d 3 x (6.89) 

Jv c 



^Pmech _ j (^ j_ 1 T v BU3, 

dt 



We have converted the sum over particles to an integral over charge and 
current densities for convenience in manipulation. The particulate nature 
can be recovered at any stage by making use of delta functions, as in 
Section 1.2. In the same manner as for Poynting's theorem, we use 
Maxwell's equations to eliminate p and J from (6.89): 



4tt 4tt\ c dt/ 



(6.90) 



192 Classical Electrodynamics 

Note that we have written only E and B in (6.90), and not H or D. The 
reason is, as mentioned earlier, that we are imagining all the charges as 
treated in the mechanical part of the system and so use the microscopic 
equations which involve only E and B. Some remarks will be made in 
the next section on the differences which arise when some of the particles, 
namely, the bound atoms, are included in the "field" energy and momentum 
through the dielectric constant and permeability. (See also Problem 6.8.) 
With (6.90) substituted into (6.89) the integrand becomes 



pE + -JxB = — 



1 /5F 

E(V • E) + - B x — - B x (V x B) 



(6.91) 



c Att\- c dt 

Then writing 

Bx^=-i( E xB) + Ex?5 

dt dt } dt 

and adding B(V • B) = to the square bracket, we obtain 

P E + - J x B = — [E(V-E) + B(V-B) - E x (V x E) 

c 4n 

- B x (V x B)] - — - (E x B) (6.92) 

4rrcdt 

The rate of change of mechanical momentum (6.89) can now be written 

^JES* + ± f J_( E XB)A = 1( [E(V-E) - E x (V x E) 
dt dtJv47TC 4ttJv 

+ B(V • B) - B x (V x B)] d*x (6.93) 

We may tentatively identify the volume integral on the left as the total 
electromagnetic momentum P fleld in the volume V: 



field 



= — (E x B) d 3 x (6.94) 

AttcJv 



The integrand can be interpreted as a density of electromagnetic 
momentum. We note that this momentum density is proportional to the 
energy-flux density S, with proportionality constant c~ 2 . , 
To complete the identification of the volume integral of — (E x B) as 

electromagnetic momentum, and to establish (6.93) as the conservation 
law for momentum, we must convert the volume integral on the right into 
a surface integral of the normal component of something which can be 
identified as momentum flow. 



[Sect. 6.9] Time-Varying Fields, Maxwell's Equations, Conservation Laws 193 

Evidently the terms in the volume integral (6.93) transform like vectors. 
Consequently, if they are to be combined into the divergence of some 
quantity, that quantity must be a tensor of the second rank. While it is 
possible to deal with rectangular components of momentum, instead of the 
vectorial form (6.93), the tensor can be handled within the framework of 
vector operations by introducing a corresponding dyadic. If a tensor in 
three dimensions is denoted by T u (i,j =1,2, 3), and e* are the unit base 
vectors of the coordinate axes, the dyadic corresponding to the tensor T i} 
is defined to be 

Y=| le^e. (6.95) 

t=l 1 = 1 

The unit vector on the left can form scalar or vector products from the 
left, and correspondingly for the unit vector on the right. Given the dyadic, 
we can determine the tensor elements by taking the appropriate scalar 
products : 

r„ = e, -V. c, (6.96) 

A special dyadic is the identity I formed with the unit second-rank tensor: 

<— > 

I = € x € x + e 2 € 2 + £,€3 (6.97) 

<-> 
The scalar product of any vector or vector operation with I from either 

the left or right merely gives the original vector quantity. 

With these sketchy remarks about dyadics, we now proceed with the 

vector manipulations needed to convert the volume integral on the right 

side of (6.93) into a surface integral. Using the vector identity, 

£V(B • B) = (B • V)B + B x (V x B) 

the terms involving B in (6.93) can be written 

B(V • B) - B x (V x B) = B(V • B) + (B • V)B - %VB 2 (6.98) 

This can be identified as the divergence of a dyadic: 

B(V . B) + (B • V)B - %VB 2 = V • (BB - YlB 2 ) (6.99) 

The electric field term in (6.93) can be put in this same form. Consequently 
the conservation of linear momentum becomes 

T (Pmech + Pfleld) = | V -Yd 8 * = in .? da (6.100) 

dt Jv Js 

<— > 
The tensor-dyadic T, called Maxwell's stress tensor, is 

Y = — [EE + BB - \\(E 2 + fl 2 )] (6.101) 

Att 



194 Classical Electrodynamics 

The elements of the tensor are 

T it = J- \E t E, + B t B t - tfifp + fl 2 )] (6.102) 

477 

Evidently (— n • T) in (6.100) represents the normal flow of momentum 
per unit area out of the volume V through the surface S. Or, in other 

words, (— n • T) is the force per unit area transmitted across surface S. 
This can be used to calculate the forces acting on material objects in 
electromagnetic fields by enclosing the objects with a boundary surface S 
and adding up the total electromagnetic force according to the right-hand 
side of (6. 100). 

The conservation of angular momentum of the combined system of 
particles and fields can be treated in the same way as we have handled 
energy and linear momentum. This is left as a problem for the student 
(Problem 6.9). 



6.10 Macroscopic Equations 

Although the equations of electrodynamics have been written in macro- 
scopic form for the most part in this chapter, the reader will be aware that 
the derivation of the macroscopic equations from the microscopic ones 
was done separately for electro statics and magnetostatics in Sections 4.3 
and 5.8. Thus there arises the question of whether the derivation still holds 
good for time-dependent fields. It is intuitively obvious that it must, since 
Maxwell's addition of the displacement current was done at the macro- 
scopic level. Nevertheless, it is useful to examine briefly the derivation to 
see in particular how the time variation of the polarization P gives rise to 
a current contribution and so converts the microscopic displacement 
current dE/dt into the macroscopic displacement current dD/dt. 

The basic assumption inherent in our previous discussions was that the 
macroscopic fields E and B which satisfy the two homogeneous Maxwell's 
equations (6.28) are the averages of the corresponding microscopic fields 
e and (3 : 

E(x,0=<€>, B(x,0 = <P> (6.103) 

The averages now involve a temporal and a spatial average, e.g., 

<e> = -g±^ jd*tjdTt(x + 5, t + t) (6-104) 

where the volume AK and the time interval Ar are small compared to 
macroscopic quantities. 



[Sect. 6.10] Time-Varying Fields, Maxwell's Equations, Conservation Laws 195 

Relations (6.103) imply that the macroscopic potentials C> and A are the 
averages of their microscopic counterparts, 

0>(x, /) = <<£>, A(x,0=<a> (6.105) 

since the fields E and B (or e and (5) are derived by differentiation according 
to (6.29) and (6.31). 

The derivation of the averaged potentials in terms of the molecular 
properties proceeds exactly as in Sections 4.3 and 5.8, with two modifi- 
cations. The first is that, according to our discussion of the solution of the 
wave equation in Section 6.6, we must have a "retarded" solution. Thus the 
same steps that led to (4.33) now lead to the averaged scalar potential: 

(<f>) = ( N (x')\^^P- + < Pmo i(x', O) ' W— L ^t)1 ** 

J L |x — X | \|X — x|/Jret 

(6.106) 
To this must be added the retarded contribution of the excess free charge, 
p cx . The second change comes in the vector potential. In the steady-state 
situation the molecular contribution to the vector potential was the sum of 
terms like (5.75), representing magnetic dipole contributions. The leading 
term in the expansion vanished because of the condition V • J = 0. With 
time-dependent fields this is no longer true. If we retrace our steps to 
equation (5.51), we see that the leading term in the expansion of a mol is 

~ ~ * = * I*-"' ° " + ("te™^) 016 ) + • • • < 61 ° 7 > 

For simplicity we omit the retarded symbols temporarily. Using the 
identity V • (x/J) = J t + x/V . J, the first term can be transformed into 



a mo i = —, x'(V' • J mo i) d 3 x' + (5.75) + 

ex — xj J 



(6.108) 



The continuity equation can now be used to write V • J mol = —dp mo Jdt', 
and the definition of molecular electric dipole moment (4.25) can be 
employed to cast (6.108) in the form: 



a mo i(x, t) 



j dp 3 (0 1 + m,.(Q x (x - x 3 )" 



(6.109) 



-C dt' |X — X,-| ' |X — X,-| 3 J ret 

With time-dependent fields we have a leading term proportional to the 

time rate of change of the electric dipole moment.* Summing over all 

* Actually, for time-varying fields not only does the leading term appear, but also from 

3Q / 1 \ <-> 

the second term in (5.51) there arises a term which is — — • V I • I , where Q 

c dt \|x - x |/ 

is the quadrupole dyadic of the molecule. Since we kept only electric dipole terms in 
(6.106), we drop this quadrupole term here. 



196 



Classical Electrodynamics 



molecules and averaging according to (6.104) leads to the averaged 
microscopic vector potential: 



(a) 



')> 



= ifjV(x')r-^--|;<Pmoi(x',n 
c J L|x — x | Ot 

+ c<m mo i(x', i')> x W , * ) dV (6.110) 

\ X — X / Jret 



To this must be added the standard contribution from the macroscopic 
conduction-current density J(x, *)• 

Solutions (6.106) and (6.110), augmented by the free-charge and con- 
duction-current contributions, can be written as 



<*>-Jfc^L" 1 

cJ L x — x 



(6.111) 



With definitions (4.36) and (5.77) of macroscopic polarization P and 
magnetization M, the averaged charge and current densities in (6.111) can 
be expressed as 

(p) = p-V-P 

ot 

where p and J are the macroscopic charge and current densities. 

We are now in a position to verify explicitly the deduction of the 
macroscopic Maxwell's equations from the microscopic ones. The 
homogeneous ones follow directly from identification (6.103). Of the 
inhomogeneous ones, consider the microscopic form of Ampere's law: 



_ _ 4t7 t , 13c 
c c ot 



(6.113) 



Averaging both sides and using (6.112) for <J>, we get 

4tt / „ 3P\ 1 9E tr „ „ x 

VxB = T (j + c(V*M) + ^)+- F (6.114) 

With the definitions H = B — 47rM and D = E + 4ttP, this becomes 

(6.115) 



_ TT 4tt t , ldD 
V x H = — J + - — 

c c ot 



as required. The other equation, V • D = Airp, follows in an even simpler 
way from (6.112). 



[Sect. 6.10] Time-Varying Fields, Maxwell's Equations, Conservation Laws 197 

As a final remark concerning the macroscopic field equations we discuss 
the differences between the microscopic and macroscopic forms of 
Poynting's theorem. We derived the conservation of energy in Section 
6.8 in the macroscopic form (6.81). Written out explicitly in terms of all 
the fields, it is 

-^f (E x H).n«fa + -f (e.^ + H—W*^ _f E.Jd*x 
Air J s AttJv\ ot at/ Jv 

(6.116) 

The different fields E, D, B, H enter in characteristic ways which can be 
understood if we establish contact with the microscopic form of Poynting's 
theorem. We can do this most easily by merely expressing the left side of 

(6.116) in terms of the basic fields E and B. Then (6.116) can easily be 
shown to be 

^((ExB).nrfa + i[ ( E • ^ + B . ??) d*x 
AttJs AttJv \ at ot I 

= - I E •( J + cV x M + — j d z x (6.117) 

From (6.112) we see that (6.117) looks like the statement of Poynting's 
theorem for the microscopic fields, except that each quantity is replaced by 
its average. This is not the average of Poynting's theorem for microscopic 
fields, but differs from it by a set of terms which are the statement of energy 
conservation for the fluctuating fields measuring the instantaneous 
departure of € and (3 from E and B. Apart from these fluctuating fields, 

(6.117) can be interpreted as follows. 

If we include in the sources of charge and current the electronic motion 
within the molecules as well as the conduction current, then Poynting's 
theorem appears in terms of the basic fields E and B and involves the work 
done per unit time by the electric field on all currents. If we choose to 

rap 1 

treat the work done on the effective molecular current — + c(V x M) 

-Ot 

as energy stored or propagated in the medium, that term can be taken 
over to the left-hand side and included in the energy-density and energy- 
flow terms characteristic of the medium. Then we return to the macro- 
scopic Poynting's theorem (6.116) with only the work done per unit time 
by the electric field on the conduction current shown explicitly. It is 
natural to absorb the energy associated with the effective molecular 
current into the energy stored in the field, since it is a property of the 
medium and is in general stored energy (i.e., reactive power) which involves 
no time-average dissipation (not true for magnetic media with hysteresis 



198 Classical Electrodynamics 

effects). The power associated with the conduction current is, on the 
other hand, dissipative, since it involves a conversion of electrical energy 
into mechanical. 



REFERENCES AND SUGGESTED READING 

The conservation laws for the energy and momentum of electromagnetic fields are 
discussed in almost all textbooks. A good treatment of the energy of quasi-stationary 
currents and forces acting on circuits carrying currents, different from ours, is given by 

Panofsky and Phillips, Chapter 10. 

Stratton, Chapter II, 
discusses the Maxwell stress tensor in some detail in considering forces in fluids and 
solids. The general topics of conservation laws, as well as quasi-stationary circuit theory, 
inductance calculations, and forces, are dealt with in a lucid manner by 

Abraham and Becker, Band I, Chapters VIII and IX. 
Inductance calculations and circuit theory are also presented by 

Smythe, Chapters VIII, IX, and X, 
and many engineering textbooks. 

The here-neglected subject of eddy currents and induction heating is discussed with 
many examples by 

Smythe, Chapter XI. 
The mathematical topics in this chapter center around the wave equation. The 
initial-value problem in one, two, three, and more dimenisons is discussed by 

Morse and Feshbach, pp. 843-847, 
and, in more mathematical detail, by 

Hadamard. 



PROBLEMS 

6.1 (a) Show that for a system of current-carrying elements in empty space the 
total energy in the magnetic field is 

w - .,,...,^J(xW(x') 



= h*r x y x ' 



where J(x) is the current density. 

(b) If the current configuration consists of n circuits carrying currents 
I x , I 2 , ■ ■ ■ , /„, show that the energy can be expressed as 

n n n 

W = \ I Ul? +2 2 Miihh 

i=l i=\ j>i 

Exhibit integral expressions for the self-inductances (L 2 ) and the mutual 
inductances (M i3 ). 
6.2 A two-wire transmission line consists of a pair of nonpermeable parallel 
wires of radii a and b separated by a distance d > a + b, A current flows 



[Probs. 6] Time- Varying Fields, Maxwell's Equations, Conservation Laws 199 

down one wire and back the other. It is uniformly distributed over the 
cross section of each wire. Show that the self-inductance per unit length is 



6.3 



6.4 



6.5 



c 2 L = 1 + 2 In 



© 



A circuit consists of a thin conducting shell of radius a and a parallel 
return wire of radius b inside. If the current is assumed distributed 
uniformly throughout the cross section of the wire, calculate the self- 
inductance per unit length. What is the self-inductance if the inner con- 
ductor is a thin hollow tube ? 

Show that the mutual inductance of two circular coaxial loops in a homo- 
geneous medium of permeability /* is 



M 12 = ^ Vab 



where 



[(H- 



(!-*)*<*)-!**) 



k 2 = 



Aab 
(a + bf + d 2 



and a, b are the radii of the loops, d is the distance between their centers, 
and K and E are the complete elliptic integrals. 

Find the limiting value when d < a, 6 and a ~ b. 
A transmission line consists of two, parallel perfect conductors of arbitrary, 
but constant, cross section. Current flows down one conductor and returns 
via the other. 




Show that the product of the inductance per unit length L and the 
capacitance per unit length C is 



LC = 



fi€ 



where /a, and e are the permeability and the dielectric constant of the medium 
surrounding the conductors, while c is the velocity of light in vacuo. 
6.6 Prove that any vector field F can be decomposed into transverse and 
longitudinal parts, 

F = F £ + ¥ t 

with V-F ( =OandVxF ( =0, where F { and F t are given by (6.49) and 
(6.50). 



200 Classical Electrodynamics 

6.7 (a) Show that the one-dimensional wave equation, 

d 2 ip 1 d^xp 

~fa?~c*~di* == ° 
has the general solution, 

I x\ I x\ c C t+ix/c) 

where the boundary conditions are specified by the values of y> and dy>/ dx 
at x = for all time: 

V'(0,0=/(0, d £(0,t)=F(t) 

(b) What is the corresponding solution if the boundary conditions are 
that, at t = 0, 

y>(x,Q)=f{x), ?l(x,0)=g(x) 

6.8 Discuss the conservation of energy and linear momentum for a macro- 
scopic system of sources and electromagnetic fields in a medium described 
by a dielectric constant e and a permeability fi. Show that the energy 
density, Poynting's vector, field-momentum density, and Maxwell stress 
tensor are given by 

u = i- (eE 2 + fxH 2 ) 

OTT 

S = -£ (E x H) 

g =£I(ExH) 

'rTT 

What modifications arise if e and fj, are functions of position ? 

6.9 With the same assumptions as in Problem 6.8 discuss the conservation of 
angular momentum. Show that the differential and integral forms of the 
conservation law are 



and 



r\ 4 v 

Y t (^mech + Afield) + V • M = 
j\ (J^mech + Afield) d 3 X + \ n • M da = 



where the field angular-momentum density is 

J^fleld = XXg=£^XX(ExH) 



[Probs. 6] Time-Varying Fields, MaxwelVs Equations, Conservation Laws 201 
and the flux of angular momentum is described by the tensor 

M = T x x 



<— >• 



Note: M can be written as a third-rank tensor, M m = T f jX k — r,-^. 
But in the indices j and it it is antisymmetric and so has only three inde- 
pendent elements. Including the index i, M m therefore has nine components 
and can be written as a pseudo tensor of the second rank, as above. 

6.10 A plane wave is incident normally on a perfectly absorbing flat screen. 

(a) From the law of conservation of linear momentum show that the 
pressure (called radiation pressure) exerted on the screen is equal to the 
field energy per unit volume in the wave. 

(b) In the neighborhood of the earth the flux of electromagnetic energy 
from the sun is approximately 0.14 watt/cm 2 . If an interplanetary "sail- 
plane" had a sail of mass 10~ 4 gm/cm 2 of area and negligible other weight, 
what would be its maximum acceleration in centimeters per square second 
due to the solar radiation pressure? How does this compare with the 
acceleration due to the solar "wind" (corpuscular radiation)? 

6.11 A circularly polarized plane wave moving in the z direction has a finite 
extent in the x and y directions. Assuming that the amplitude modulation 
is slowly varying (the wave is many wavelengths broad), show that the 
electric and magnetic fields are given approximately by 



r i(*E a dE a \ 

E(x, y, z, t) =* \E (x, yfa ± fc») + j\~^ ± i ~^J^ 



gikz—icot 



B=;T iVyue E 
where e x , e 2 , e 3 are unit vectors in the x, y, z directions. 
6.12 For the circularly polarized wave of Problem 6.11 calculate the time- 
averaged component of angular momentum parallel to the direction of 
propagation. Show that the ratio of this component of angular momentum 
to the energy of the wave is 

u 

Interpret this result in terms of quanta of radiation (photons). Show that 
for a cylindrically symmetric, finite plane wave the transverse components 
of angular momentum vanish. 



7 



Plane Electromagnetic Waves 



This chapter is concerned with plane waves in unbounded, or 
perhaps semif infinite, media. The basic properties of plane waves in non- 
conducting media — their transverse nature, the various states of polari- 
zation — are treated first. Then the behavior of one-dimensional wave 
packets is discussed; group velocity is introduced; dispersive effects are 
considered. Reflection and refraction of radiation at a plane interface 
between dielectrics are presented. Then plane waves in a conducting 
medium are described, and a simple model of electrical conductivity is 
discussed. Finally the conductivity model is modified to apply to a 
tenuous plasma, or electron gas, and the propagation of transverse waves 
in a plasma in the presence of an external static magnetic field is con- 
sidered. 



7.1 Plane Waves in a Nonconducting Medium 

A basic feature of Maxwell's equations for the electromagnetic field is 
the existence of traveling wave solutions which represent the transport of 
energy from one point to another. The simplest and most fundamental 
electromagnetic waves are transverse, plane waves. We proceed to see how 
such solutions can be obtained in simple nonconducting media described 
by spatially constant permeability and susceptibility. In the absence of 
sources, Maxwell's equations in an infinite medium are : 



V-E = V x E + -— =0 

c dt 

V-B = VxB-^^ = 

c dt 

202 



(7.1) 



[Sect. 7.1] Plane Electromagnetic Waves 203 

where the medium is characterized by the parameters fi, e. By combining 
the two curl equations and making use of the vanishing divergences, we 
find easily that each cartesian component of E and B satisfies the wave 
equation: 

V2 "-I?i = (7 - 2) 

v 2 dt 2 



where 



_ c 



(7.3) 



is a constant of the dimensions of velocity characteristic of the medium. 
The wave equation (7.2) has the well-known plane-wave solutions: 



k-x-i<ot /n ^\ 



u = e 

where the frequency co and the magnitude of the wave vector k are related 
by 



CO , — CO 



fc = ^ = V^ (7.5) 

V c 

If we consider waves propagating in only one direction, say, the x 
direction, the fundamental solution is 

u(x, i) = Ae ikx - iat + Be- ikx - imt (7.6) 

Using (7.5), this can be written 

u k (x, t) = Ae ik <*- Vt) + Be~ mx+Vt) (7.7) 

If v is not a function of A: (i.e., a nondispersive medium, with fxe independent 
of frequency), we know by the Fourier integral theorem (2.50) and (2.51) 
that by linear superposition we can construct from u k (x, t) a general 
solution of the form: 

u (x, t) =f(x - vt) + g(x + vt) (7.8) 

where f(z) and g(z) are arbitrary functions. It is easy to verify directly 
that this is a solution of the wave equation (7.2). Equation (7.8) represents 
waves traveling to the right and to the left with velocities of propagation 
equal to v, which is called the phase velocity of the wave. If v is a function 
of k, the situation is not as simple— the initial waves f(x) and g(x) are not 
propagated without distortion at velocity v (see Section 7.3). For each 
frequency component, however, v given by (7.3) is still the phase velocity. 
The basic plane wave (7.4) and (7.5) satisfies the scalar-wave equation 
(7.2). But we still must consider the vector nature of the electromagnetic 
fields and the requirement of satisfying Maxwell's equations. With the 
convention that the physical electric and magnetic fields are obtained by 



204 



Classical Electrodynamics 



taking the real parts of complex quantities, we assume that the plane- 
wave fields are of the form : 



E(x,t) = e 1 E e*-*- ia * 

ik • x—i<ot 



(7.9) 



B(x, = ejV 

where c l5 c 2 are two constant real unit vectors, and E , B are complex 
amplitudes which are constant in space and time. The requirements 
V • E = and V • B = demand that 



€l • k = 0, 



c 2 • k = 



(7.10) 



This means that E and B are both perpendicular to the direction of 
propagation k. Such a wave is called a transverse wave. The curl equations 
provide further restrictions. Substitution of (7.9) into the first curl 
equation in (7.1) leads to the relation: 



(k x €!)£ €380 



ik • x— i<"t 



= 



Equation (7.11) (really several equations) has the solution: 



€„ = 



kxc t 



and 



B = V^e E 



(7.11) 

(7.12) 
(7.13) 



This shows that (c l5 c 2 , k) form a set of orthogonal vectors and that E and 
B are in phase and in constant ratio, as indicated in Fig. 7.1. The wave 
described by (7.9), (7.12), and (7.13) is a transverse wave propagating in 
the direction k. It represents a time-averaged flux of energy given by the 




Fig. 7.1 Propagation vector k and 

two orthogonal polarization vectors 

6i and € 8 . 



[Sect. 7.2] Plane Electromagnetic Waves 205 

real part of the complex Poynting's vector: 

S = -— E x H* (7.14) 

2 4tt 

The energy flow (energy per unit area per unit time) is 

S = ^ /^|£ | 2 € 3 (7.15) 



J 0\ «3 
87T 'V fl 

where €3 is a unit vector in the direction of k. The time-averaged density 
u is correspondingly 

u = — (cE • E* + -B • B*) (7.16) 

l&TT fl 

This gives 

M = f|£ol 2 (7-17) 

The ratio of the magnitude of (7.15) to (7.17) shows that the velocity of 
energy flow is v = clV/txe, as expected from (7.8). 



7.2 Linear and Circular Polarization 

The plane wave (7.9) is a wave with its electric field vector always in the 
direction e v Such a wave is said to be linearly polarized with polarization 
vector € v To describe a general state of polarization we need another 
linearly polarized wave which is independent of the first. Clearly the two 
waves 






ik • x— io>t 



with B, = y/fte — - — '-, j = 1,2 



(7.18) 



represent two such linearly independent solutions. The amplitudes E x 
and E 2 are complex numbers to allow the possibility of a phase difference 
between the waves. A general solution for a plane wave propagating in 
the direction k is given by a linear combination of E x and E 2 : 

E(x, = (eA + c^y* * x - i(ot (7.19) 

If E x and E 2 have the same phase, (7.19) represents a linearly polarized 
wave, with its polarization vector making an angle 6 = tan -1 (EJE-^) with 
c x and a magnitude E = Ve^ + E 2 2 , as shown in Fig. 7.2. 

If E x andE 2 have different phases, the wave (7.19) is elliptically polarized. 




Classical Electrodynamics 
E 



Fig. 7.2 Electric field of a linearly polarized 
wave. 



To understand what this means let us consider the simplest case, circular 
polarization. Then E x and E 2 have the same magnitude, but differ in phase 
by 90°. The wave (7.19) becomes: 

E(x, = £ ( ei ± /e 2 y' k • % - iait (7.20) 

with E the common real amplitude. We imagine axes chosen so that the 
wave is propagating in the positive z direction, while e x and € 2 are in the x 
and y directions, respectively. Then the components of the actual electric 
field, obtained by taking the real part of (7.20), are 



(7.21) 



EJx, t) = E cos (kz — cut) 
E y (x, t) = TE sin (kz — wt) 

At a fixed point in space, the fields (7.21) are such that the electric vector 
is constant in magnitude, but sweeps around in a circle at a frequency co, 
as shown in Fig. 7.3. For the upper sign (e x + /e 2 ), the rotation is counter- 
clockwise when the observer is facing into the oncoming wave. This wave 
is called left circularly polarized in optics. In the terminology of modern 
physics, however, such a wave is said to have positive helicity. The latter 
description seems more appropriate because such a wave has a positive 
projection of angular momentum on the z axis (see Problem 6.12). For 
the lower sign (c x — /e 2 ), the rotation of E is clockwise when looking into 




E(x, = i5o(ci + i«2)«' 



Fig. 7.3 Electric field of a circularly polarized 
wave. 



[Sect. 7.2] 



Plane Electromagnetic Waves 



207 




Fig. 7.4 Electric field and magnetic induction for an elliptically polarized wave. 

the wave; the wave is right circularly polarized (optics); it has negative 
helicity. 

The two circularly polarized waves (7.20) form an equally acceptable 
set of basic fields for description of a general state of polarization. We 
introduce the complex orthogonal unit vectors : 



with properties 



c± = -7= (ci ± ie 2 ) 



(7.22) 



(7.23) 



(7.24) 



e ± *.€ T =0 

e±*-e 3 = 

€ ± *.€ ± =1 

Then a general representation, equivalent to (7.19), is 

E(x, t) = (£+€+ + E_€_)e 

where E+ and E_ are complex amplitudes. If E + and E_ have different 
magnitudes, but the same phase, (7.24) represents an elliptically polarized 
wave with principal axes of the ellipse in the directions of e x and c 2 . The 
ratio of semimajor to semiminor axis is (1 + r)/(l — r), where EJE + = r. 
If the amplitudes have a phase difference between them, E_/E + = re m , 
then it is easy to show that the ellipse traced out by the E vector has its 
axes rotated by an angle (a/2). Figure 7.4 shows the general case of elliptical 
polarization and the ellipses traced out by both E and B at a given point in 
space. 

For r = ± 1 we get back a linearly polarized wave. 



208 Classical Electrodynamics 

7.3 Superposition of Waves in One Dimension; Group Velocity 

In the previous sections plane-wave solutions to Maxwell's equations 
were found and their properties discussed. Only monochromatic waves, 
those with a definite frequency and wave number, were treated. In actual 
circumstances such idealized solutions do not arise. Even in the most 
monochromatic light source or the most sharply tuned radio transmitter 
or receiver, one deals with a finite (although perhaps small) spread of 
frequencies or wavelengths. This spread may originate in the finite 
duration of a pulse, in inherent broadening in the source, or in many 
other ways. Since the basic equations are linear, it is in principle an 
elementary matter to make the appropriate linear superposition of 
solutions with different frequencies. In general, however, there are several 
new features which arise. 

1. If the medium is dispersive (i.e., the dielectric constant is a function 
of the frequency of the fields), the phase velocity is not the same for each 
frequency component of the wave. Consequently different components of 
the wave travel with different speeds and tend to change phase with respect 
to one another. This leads to a change in the shape of a pulse, for example, 
as it travels along. 

2. In a dispersive medium the velocity of energy flow may differ greatly 
from the phase velocity, or may even lack precise meaning. 

3. In a dissipative medium, a pulse of radiation will be attenuated as it 
travels with or without distortion, depending on whether the dissipative 
effects are or are not sensitive functions of frequency. 

The essentials of these dispersive and dissipative effects are implicit in 
the ideas of Fourier series and integrals (Section 2.9). For simplicity, we 
consider scalar waves in only one dimension. The scalar amplitude 
u(x, i) can be thought of as one of the components of the electromagnetic 
field. The basic solution to the wave equation (7.2) has been exhibited in 
(7.6). The relationship between frequency w and wave number k is given 
by (7.5) for the electromagnetic field. Either co or k can be viewed as the 
independent variable when one considers making a linear superposition. 
Initially we will find it most convenient to use k as an independent variable. 
To allow for the possibility of dispersion we will consider co as a general 
function of k: 

co = co(k) (7.25) 

Since the dispersive properties cannot depend on whether the wave travels 
to the left or to the right, co must be an even function of k, co( —k) = 
oj(k). For most wavelengths co is a smoothly varying function of A:. But at 



[Sect. 7.3] Plane Electromagnetic Waves 209 

certain frequencies there are regions of "anomalous dispersion" where to 
varies rapidly over a narrow interval of wavelengths. With the general form 
(7.25), our subsequent discussion can apply equally well to electromagnetic 
waves, sound waves, de Broglie matter waves, etc. For the present we 
assume that k and <o{k) are real, and so exclude dissipative effects. 

From the basic solutions (7.6) we can build up a general solution of the 
form 

u(x, i) = -4= 1*°° A(k)e ikx - it0(k)t dk (7.26) 

The factor I/VItt has been inserted to conform with the Fourier integral 
notation of (2.50) and (2.51). The amplitude A(k) describes the properties 
of the linear superposition of the different waves. It is given by the 
transform of the spatial amplitude u(x, t), evaluated at t = 0*: 

A(k) = -j= f °° u(x, 0)e- ikx dx (7.27) 

J2TT J -co 

If u(x, 0) represents a harmonic wave e ik » x for all x, the orthogonality 
relation (2.52) shows that A(k) = V2^d(k — k ), corresponding to a 
monochromatic traveling wave u(x, i) = e^"^ as required. If, 
however, at t = 0, u(x, 0) represents a finite wave train with a length of 
order Ax, as shown in Fig. 7.5, then the amplitude A(k) is not a delta 
function. Rather, it is a peaked function with a breadth of the order of Ak, 
centered around a wave number k which is the dominant wave number in 
the modulated wave u(x, 0). If Ax and Ak are defined as the rms deviations 
from the average values of x and k [defined in terms of the intensities 
\u(x, 0)| 2 and |i4(fc)| 2 ], it is possible to draw the general conclusion: 

AxAk>% (7.28) 

The reader may readily verify that, for most reasonable pulses or wave 
packets which do not cut off too violently, Ax times Ak lies near the lower 
limiting value in (7.28). This means that short wave trains with only a 
few wavelengths present have a very wide distribution of wave numbers of 
monochromatic waves, and conversely that long sinusoidal wave trains 
are almost monochromatic. Relation (7.28) applies equally well to 
distributions in time and in frequency. 
The next question is the behavior of a pulse or finite wave train in time. 

* The following discussion slights somewhat the initial-value problem. For a second- 
order differential equation we must specify not only u(x, 0) but also du(x, 0)1 dt. This 
omission is of no consequence for the rest of the material in this section. It is remedied in 
the following section. 



210 



u(x, 0) 



Classical Electrodynamics 




A(k) 



Fig. 7.5 A harmonic wave train 

of finite extent and its Fourier 

spectrum in wave number. 

The pulse shown at t = in Fig. 7.5 begins to move as time goes on. The 
different frequency or wave-number components in it move at different 
phase velocities. Consequently there is a tendency for the original 
coherence to be lost and for the pulse to become distorted in shape. At the 
very least, we might expect it to propagate with a rather different velocity 
from, say, the average phase velocity of its component waves. The general 
case of a highly dispersive medium or a very sharp pulse with a great 
spread of wave numbers present is difficult to treat. But the propagation 
of a pulse which is not too broad in its wave-number spectrum, or a pulse 
in a medium for which the frequency depends weakly on wave number, can 
be handled in the following approximate way. The wave at time t is given 
by (7.26). If the distribution A(k) is fairly sharply peaked around some 
value k , then the frequency oy{k) can be expanded around that value of k: 



a)(k) = w + — 
dk 



(k - k ) + 



(7.29) 



and the integral performed. Thus 

e ilk (d<o/dk)\ -(o ]t /*oo 

u(x, t) ~ — ,4(fe) e «*-(*»/*)Io*]* dk (7 30) 

yJllT J -oo 

From (7.27) and its inverse it is apparent that the integral in (7.30) is just 
u{x', 0), where x' = x — (da>ldk)\ t: 

u(x, t)~ ulx — t, o)e i[ ' Co(< * to/< ** ) lo- £,, o]* / 7 2i) 

V dk o / 



[Sect. 7.3] Plane Electromagnetic Waves 211 

This shows that, apart from an overall phase factor, the pulse travels along 
undistorted in shape with a velocity, called the group velocity: 



dco 



(7.32) 



If an energy density is associated with the magnitude of the wave (or its 
absolute square), it is clear that in this approximation the transport of 
energy occurs with the group velocity, since that is the rate at which the 
pulse travels along. 

For light waves the relation between co and k is given by 

a>(k) = -%- (7.33) 

n(k) 

where c is the velocity of light in vacuum, and n(k) is the index of refraction 
expressed as a function of k. The phase velocity is 

,, = ^ = -S- (7.34) 

k n{k) 

and is greater or smaller than c depending on whether n{k) is smaller or 
larger than unity. For most optical wavelengths n{k) is greater than unity 
in almost all substances. The group velocity (7.32) is 



v a = (7.35) 

\n(co) + co(dn/dco)] 

In this equation it is more convenient to think of n as a function of co than 
of k. For normal dispersion (dn/dco) > 0, and also n > 1 ; then the 
velocity of energy flow is less than the phase velocity and also less than c. 
In regions of anomalous dispersion, however, dn/dco can become large and 
negative. Then the group velocity differs greatly from the phase velocity, 
often becoming larger than c. * The behavior of group and phase velocities 
as a function of frequency in the neighborhood of a region of anomalous 
dispersion is shown in Fig. 7.6. 

* There is no cause for alarm that our ideas of special relativity are here violated; 
group velocity is no longer a meaningful concept. A large value of dn/dco is equivalent to 
a rapid variation of eo as a function of k. Consequently the approximations made in 
(7.29) ff. are no longer valid. The behavior of the pulse is much more involved. 



212 



Classical Electrodynamics 



n (go) 




Fig. 7.6 Index of refraction n(co) 
as a function of frequency eo at 
a region of anomalous disper- 
sion ; phase velocity v„ and group 
velocity v g as functions of to. 



7.4 Illustration of Propagation of a Pulse in a Dispersive Medium 

To illustrate the ideas of the previous section and to show the validity 
of the concept of group velocity we will now consider a specific model for 
the dependence of frequency on wave number and will calculate without 
approximations the propagation of a pulse in this model medium. Before 
specifying the particular model it is necessary to state the initial-value 
problem in more detail than was done in (7.26) and (7.27). As noted there, 
the proper specification of an initial- value problem for the wave equation 
demands the initial values of both function u(x, 0) and time derivative 
du(x, 0)/dt. If we agree to take the real part of (7.26) to obtain u(x, t), 

i i r°° 

u(x, = - -== A (k)e ikx - i(oWt dk + c. c. (7.36) 

then it is easy to show that A(k) is given in terms of the initial values by: 
_ 1_ I °° -ihx, / twx . i du 

J2.TT J -oo 



A(k) 



u(x,0) + 



(*,0) 



dx (7.37) 



co(k) dt 
We will take a Gaussian modulated oscillation 

u(x, 0) = e ~ x2/2L * cos k x (7.38) 

as the initial shape of the pulse. For simplicity, we will assume that 

— (x, 0) = (7.39) 



[Sect. 7.4] Plane Electromagnetic Waves 213 

This means that at times immediately before t = the wave consisted of 
two pulses, both moving towards the origin, such that at t = they 
coalesced into the shape given by (7.38). Clearly at later times we expect 
each pulse to re-emerge on the other side of the origin. Consequently the 
initial distribution (7.38) may be expected to split into two identical 
packets, one moving to the left and one to the right. The Fourier amplitude 
A(£) for the pulse described by (7.38) and (7.39) is: 

A (fy = _4= °° e -ikx e -x*/2L 2 cos koX dx 

JItT J- oo 



Lr e -(L 2 /2)(k-k ) 2 _j_ e -(i 2 /2)(fc + fc ) 2 -j (740) 



The symmetry A(—k) = A(k) is a reflection of the presence of two pulses 
traveling away from the origin, as will be seen below. 

In order to calculate the wave form at later times we must specify 
o = a)(k). As a model allowing exact calculation and showing the 
essential dispersive effects, we assume 

co(k) = v{l + — ) (7-41) 

where v is a constant frequency, and a is a constant length which is a typical 
wavelength where dispersive effects become important. Since the pulse 
(7.38) is a modulated wave of wave number k = k , the approximate 
arguments of the preceding section imply that the two pulses will travel 
with the group velocity 

v g = ^(k ) = va"k (7.42) 

dk 

and will be essentially unaltered in shape provided the pulse is not too 
narrow in space. 

The exact behavior of the wave as a function of time is given by (7.36), 
with (7.40) for A(k)\ 

u ( x t \ = L _ Ref 00 r e -(L 2 /2)Vc-k Q ) 2 + e -(£ 2 /2)(fc + k ) 2 y**-iv<[l + (aV/2)] rffc 
lyJllT J- oo 

(7.43) 



214 



Classical Electrodynamics 



The integrals can be performed by appropriately completing the squares 
in the exponents. The result is 



u(x, i) = | Re 



exp 



(x — va 2 k t) 2 



(' + f-1- 



2L 2 1 + 



(i + ^J 



x exp 



jfc a; — iv\ 1 H =i- 1 r 



+ (h^-k Q ) 



(7.44) 
Equation (7.44) represents two pulses traveling in opposite directions. 
The peak amplitude of each pulse travels with the group velocity (7.42), 
while the modulation envelope remains Gaussian in shape. The width of 
the Gaussian is not constant, however, but increases with time. The width 
of the envelope is 



L(r) = 



L 2 + 



m 



(7.45) 



Thus the dispersive effects on the pulse are greater (for a given elapsed 
time), the sharper the envelope. The criterion for a small change in shape 
is that L > a. Of course, at long times the width of the Gaussian increases 
linearly with time 



L 



(7.46) 



but the time of attainment of this asymptotic form depends on the ratio 
(L/d). A measure of how rapidly the pulse spreads is provided by a com- 
parison of L(t) given by (7.45), with v g t = va 2 k t. Figure 7.7 shows two 
examples of curves of the position of peak amplitude (v g t) and the positions 
v g t ± L(t), which indicate the spread of the pulse, as functions of time. On 
the left the pulse is not too narrow compared to the wavelength k ~ x and 
so does not spread too rapidly. The pulse on the right, however, is so 
narrow initially that it is very rapidly spread out and scarcely represents a 
pulse after a short time. 

Although the above results have been derived for a special choice (7.38) 
of initial pulse shape and dispersion relation (7.41), their implications are 
of a more general nature. We have seen in Section 7.3 that the average 
velocity of a pulse is the group velocity v g = da>/dk = co'. The spreading 
of the pulse can be accounted for by noting that a pulse with an initial 



[Sect. 7.4] 



Plane Electromagnetic Waves 



215 



koL<\ 




Fig. 7.7 Change in shape of a wave packet as it travels along. The broad packet, 

containing many wavelengths (k„L > 1), is distorted comparatively little, while the 

narrow packet (k L < 1) broadens and diffuses out rapidly. 

spatial width A:r must have inherent in it a spread of wave numbers 
Afc ~ (1/A» ). This means that the group velocity, when evaluated for 
various k values within the pulse, has a spread in it of the order 



Ap a 



A#„ 



(7.47) 



At a time t this implies a spread in position of the order of Aty. If we 
combine the uncertainties in position by taking the square root of the sum 
of squares, we obtain the width Asc(0 at time t : 



A^^VcA^+g) 2 



(7.48) 



We note that (7.48) agrees exactly with (7.45) if we put Aa; = L. The 
expression (7.48) for Arc(0 shows the general result that, if co" =£ 0, a 
narrow pulse spreads rapidly because of its broad spectrum of wave 
numbers, and vice versa. All these ideas carry over immediately into wave 
mechanics. They form the basis of the Heisenberg uncertainty principle. 
In wave mechanics, the frequency is identified with energy divided by 
Planck's constant, while wave number is momentum divided by Planck's 
constant. 

The problem of wave packets in a dissipative, as well as dispersive, 
medium is rather complicated. Certain aspects can be discussed analyti- 
cally, but the analytical expressions are not readily interpreted physically. 
Wave packets are attenuated and distorted appreciably as they propagate. 
The reader may refer to Stratton, pp. 301-309, for a discussion of the 
problem, including numerical examples. 



216 



Classical Electrodynamics 



7.5 Reflection and Refraction of Electromagnetic Waves at a Plane 
Interface between Dielectrics 

The reflection and refraction of light at a plane surface between two 
media of different dielectric properties are familiar phenomena. The 
various aspects of the phenomena divide themselves into two classes. 

(1) Kinematic properties : 

(a) Angle of reflection equals angle of incidence. 

sin i ri 

(b) SnelPs law : — — = — , where i, r are the angles of incidence 

sin r n 

and refraction, while n, n' are the corresponding indices of re- 
fraction. 

(2) Dynamic properties : 

(a) Intensities of reflected and refracted radiation. 

(b) Phase changes and polarization. 

The kinematic properties follow immediately from the wave nature of 
the phenomena and the fact that there are boundary conditions to be 
satisfied. But they do not depend on the nature of the waves or the 
boundary conditions. On the other hand, the dynamic properties depend 
entirely on the specific nature of electromagnetic fields and their boundary 
conditions. 

The coordinate system and symbols appropriate to the problem are 
shown in Fig. 7.8. The media below and above the plane 2 = have 
permeabilities and dielectric constants //, e and (x ', e, respectively. A plane 
wave with wave vector k and frequency co is incident from medium fi, e. 
The refracted and reflected waves have wave vectors k' and k", respectively, 
and n is a unit normal directed from medium [i, e into medium [x, e'. 




Fig. 7.8 Incident wave k strikes 
plane interface between different 
media, giving rise to a reflected 
wave k" and a refracted wave k'. 



[Sect. 7.5] Plane Electromagnetic Waves 217 

According to (7.18), the three waves are 

INCIDENT 

E = E e ik - x - iwt ) 

B-VF — (? - 49) 



REFRACTED 



REFLECTED 



e' = E y k "*" ia * ) 



E" = E V k "- x - ta " ) 

— k" x E" (7.51) 



B" = yltu 



k 
The wave numbers have the magnitudes 

CO / — 

c 

CO 

c 



ik| = |k"i = /c = -v /Me 

c 

ik'i = k' = - v^ 



(7.52) 



The existence of boundary conditions at z = 0, which boundary 
conditions must be satisfied at all points on the plane ,at all times, implies 
that the spatial (and time) variation of all fields must be the same at z = 0. 
Consequently, we must have the phase factors all equal at z = 0, 

(k • x) 2=0 = (k' . x), =0 = (k" • x) z=0 (7.53) 

independent of the nature of the boundary conditions. Equation (7.53) 
contains the kinematic aspects of reflection and refraction. We see 
immediately that all three wave vectors must lie in a plane. Furthermore, 
in the notation of Fig. 7.8, 

k sin / = k' sin r = k" sin r' (7.54) 

Since k" = k, we find i = r'; the angle of incidence equals the angle of 
reflection. SnelFs law is 

!"Li = ^' = /^-'=- (7.55) 

sinr k A* e n 

The dynamic properties are contained in the boundary conditions — 
normal components of D and B are continuous; tangential components of 



218 



Classical Electrodynamics 




Fig. 7.9 Reflection and refraction 

with polarization perpendicular to 

the plane of incidence. 



E and H are continuous. In terms of fields (7.49)-(7.51) these boundary 
conditions at z = are : 

[ e (E + E ")-e'E '].n = 0l 

[k x E + k" x E " - k' x E '] • n = 

(E + E " - E ') x n = 



- (k x E + k" x E ") - - f (k' x E ') 



x n = 



(7.56) 



In applying these boundary conditions it is convenient to consider two 
separate situations, one in which the incident plane wave is linearly 
polarized with its polarization vector perpendicular to the plane of 
incidence (the plane defined by k and n), and the other in which the 
polarization vector is parallel to the plane of incidence. The general case 
of arbitrary elliptic polarization can be obtained by appropriate linear 
combinations of the two results, following the methods of Section 7.2. 

We first consider the electric field perpendicular to the plane of incidence, 
as shown in Fig. 7.9. All the electric fields are shown directed away from 
the viewer. The orientations of the B vectors are chosen to give a positive 
flow of energy in the direction of the wave vectors. Since the electric 
fields are all parallel to the surface, the first boundary condition in (7.56) 
yields nothing. The third and fourth equations in (7.56) give 



^o + Eq" ~ Eq = 



- (#o - V) cos i-J — E o cos r = 



(7.57) 



[Sect. 7.5] 



Plane Electromagnetic Waves 



219 



while the second, using Snell's law, duplicates the third. The relative 
amplitudes of the refracted and reflected waves can be found from (7.57). 
These are: 

E PERPENDICULAR TO PLANE OF INCIDENCE 



E n ' 



1 + 



E" 



1 - 



ft' tan r 

ju tan / 
ft' tan r 



1 + 



ju tan i 
ft' tan r 



2 cOs i sin r 



ft tan i sin (/ + r) 



sin (t — r) 
sin (i + r) 



(7.58) 



The expression on the right in each case is the result appropriate for 
[i' = fi,, as is generally true for optical frequencies. 

If the electric field is parallel to the plane of incidence, as shown in Fig. 
7.10, the boundary conditions involved are normal D, tangential E, and 
tangential H [the first, third, and fourth equations in (7.56)]. The 
tangential E and H continuous demand that 



cos i (E — Eq) — cos r E ' = 
1(E + E ")-Jle '=0 

fX * ft' 



(7.59) 



Normal D continuous, plus Snell's law, merely duplicates the second of 
these equations. The relative amplitudes of refracted and reflected fields 
are therefore 



Fig. 7.10 Reflection and refrac- 
tion with polarization parallel to 
the plane of incidence. 











E' 




k' 






n> 






&B' 




MV 














tit 


E 


k / 
/ B 


i \ 
B 


\ 







220 Classical Electrodynamics 

E PARALLEL TO PLANE OF INCIDENCE 



E ' 



= 2 



/* 



/U€ 



sin 1i 



2 cos i sin r 



we' ^ . sin (i + r) cos (i — r) 

^ sin 2r + — sin 2i ' 



„ „ ~ sin 2/ — sin 2r 



tan (i — r) 

E . fj, . . tan (i + r) 

sin 2r + — sin 2i 

Again the results on the right apply for fi = fi. 

For normal incidence (/ = 0), both (7.58) and (7.60) reduce to 



(7.60) 



E n ' 



In 



Eo 


it* -i 


n' + n 


E " ' 


v fx'e 


n' — n 


E 


M+i 


n' + n 



(7.61) 



For the reflected wave the sign convention is that for polarization parallel 
to the plane of incidence. This means that if ri > n there is a phase 
reversal for the reflected wave. 



7.6 Polarization by Reflection and Total Internal Reflection 

Two aspects of the dynamical relations on reflection and refraction are 
worthy of mention. The first is that for polarization parallel to the plane 
of incidence there is an angle of incidence, called Brewster's angle, for 
which there is no reflected wave. Putting p = /u for simplicity, we see 
from (7.60) that there will be no reflected wave when i + r = ttJ2. From 
Snell's law (7.55) we find that this specifies Brewster's angle to be 



in — tan x 



o 



(7.62) 



For a typical ratio (n'jn) = 1.5, i B ~ 56°. If a plane wave of mixed 
polarization is incident on a plane interface at the Brewster angle, the 
reflected radiation is completely plane polarized with polarization vector 
perpendicular to the plane of incidence. This behavior can be utilized to 



[Sect. 7.6] Plane Electromagnetic Waves 221 

produce beams of plane-polarized light, but is not as efficient as other 
means employing anisotropic properties of some dielectric media. Even 
if the unpolarized wave is reflected at angles other than the Brewster angle, 
there is a tendency for the reflected wave to be predominantly polarized 
perpendicular to the plane of incidence. The success of dark glasses which 
selectively transmit only one direction of polarization depends on this fact. 
In the domain of radiofrequencies, receiving antennas can be so oriented 
as to discriminate against surface-reflected waves (and also waves reflected 
from the ionosphere) in favor of the directly transmitted wave. 

The second phenomenon is called total internal reflection. The word 
internal implies that the incident and reflected waves are in a medium of 
larger index of refraction than the refracted wave (n > n'). Snell's law 
(7.55) shows that, if n > «', then r > i. Consequently, r = tt\2 when 
i = i , where 



-) 

n! 



; = sin" 1 (-j (7.63) 

For waves incident at / = / , the refracted wave is propagated parallel to 
the surface. There can be no energy flow across the surface. Hence at 
that angle of incidence there must be total reflection. What happens if 
/ > i ? To answer this we first note that, for i > /„, sin r > 1 . This means 
that r is a complex angle with a purely imaginary cosine. 



sin 
cos r = i 



5«Li)*_i (7.64) 

sin iJ 



sin i 

The meaning of these complex quantities becomes clear when we consider 
the propagation factor for the refracted wave : 

iW- x ik'(x sin r+z cos r) — A'[(sin i/sln i ) 2 — l]^z ik' (sin i/ sin i )x (1 f*S\ 

This shows that, for / > i , the refracted wave is propagated only parallel 
to the surface and is attenuated exponentially beyond the interface. The 
attenuation occurs within a very few wavelengths of the boundary, except 
for i ^ /„. 

Even though fields exist on the other side of the surface it is clear that 
there is no energy flow through the surface. Hence total internal reflection 
occurs for / > i . The lack of energy flow can be verified by calculating 
the time-averaged normal component of the Poynting's vector just inside 
the surface: 

S • n = — Re [n • (E' x H'*)] (7.66) 



222 Classical Electrodynamics 



c 



with H' = — (k' x E'), we find 

JU CO 

S • n = ^£- t Re [(n . k') |E '| 2 ] (7.67) 

But n • k' = k' cos r is purely imaginary, so that S • n = 0. 

The phenomenon of total internal reflection is exploited in many 
applications where it is required to transmit light without loss in intensity. 
In nuclear physics Lucite or other plastic "light pipes" are used to carry 
light emitted from a scintillation crystal because of the passage of an 
ionizing particle to a photomultiplier tube, where it is converted into a 
useful electric signal. The photomultiplier must often be some distance 
away from the scintillation crystal because of space limitations or magnetic 
fields which disturb its performance. If the light pipe is large in cross 
section compared to a wavelength of the radiation involved, the con- 
siderations presented here for a plane interface have approximate validity. 
When the dielectric medium has cross-sectional dimensions of the order 
of a wavelength, however, the precise geometry must be taken into account. 
Then the propagation is that of a dielectric wave guide (see Section 8.8). 



7.7 Waves in a Conducting Medium 

If the medium in which waves are propagating is a conductor, there are 
characteristic differences in the propagation, when compared with non- 
conducting media. If the medium is characterized by a conductivity a, 
as well as a dielectric constant e and permeability /u, Maxwell's equations 
are supplemented by Ohm's law : 

J = <rE (7.68) 

Hence they take the form : 



V./*H = VxE + ^— =0 

c dt 

V.eE = V xH--^-^E = 

c dt c 



(7.69) 



In the insulating dielectric we found that the time-varying fields were 
transverse, i.e., the field vectors E and H were perpendicular to the 
direction in which the spatial variation occurred. In the limit of zero 
frequency we know from our study of electro- and magnetostatics that the 
static fields in a dielectric are longitudinal, in the sense that the fields are 
derivable from scalar potentials and so point in the direction of the spatial 
variation. 



[Sect. 7.7] 



Plane Electromagnetic Waves 



223 



If the conductivity is not zero, modifications arise. For simplicity, 
consider fields which vary in only one spatial variable, |. We decompose 
the fields into longitudinal and transverse parts : 



E(£, = E long (|, + E tr (|, 
H(£, = H long (£, + H tr (£, j 



(7.70) 



Then, because of the properties of curl operation, we find that the trans- 
verse parts of E and H satisfy the two curl equations in (7.69), leading to 
transverse waves (see below), while the longitudinal parts satisfy the 
equations : 



ilong __ q 

31 



long 



dt 



dE 



long 



dS 



= 



/ 3 477(7 

\dt e 



Eiong = 



(7.71) 



The first pair of equations shows that the only longitudinal magnetic field 
possible is a static uniform field. This is the same situation as in an 
insulator. But the second pair in (7.71) shows that the longitudinal 
electric field is uniform in space, while having the time variation : 



£iong(l, = E e-^ t/€ 



(7.72) 



Consequently, no static longitudinal fields can exist in a conducting 
medium in the absence of an applied current density. For good conductors 
like copper, a ~ 10 17 sec -1 so that disturbances are damped out in an 
extremely short time. 

We now consider the transverse fields in the conducting medium. 
Assuming that the fields vary as exp (/k • x — icot), the first curl equation 
of (7.69) yields: 

H = — (k x E) (7.73) 

flOO 

while the second gives 



.,, __. , . CO _, 477(7 _ _ 

i(k x H) + ic — E E = 

c c 



(7.74) 



Elimination of either H or E from this last equation with (7.73) yields 



; 2 ( co 2 



+ 4777 



jua>a\ 






= 



This means that the propagation vector k is complex : 



fc 2 = [it 



\ coe I 



(7.75) 



(7.76) 



224 



Classical Electrodynamics 



The first term corresponds to the displacement-current and the second to 
the conduction-current contribution. In taking the square root to find k 
the branch is chosen to give the familiar results when a = 0. Then one 
finds, assuming that a is real, 



where 



k = a + ifl 



a / — co 



1 + 



(£)' 



± 1 



(7.77) 



(477(7 \ 
< 1 I we find approximately 



k = a + iff — yjfxe \- i 



Itt ///, 



(7.78) 



correct to first order in (a/coe). In this limit Re A: > Im k and the attenua- 
tion of the wave (Im k) is independent of frequency, aside from the possible 

frequency variation of the conductivity. For a good conductor I > 1 I, 

on the other hand, a and /? are approximately equal: v (xi€ 



k ^ (I + /) ^ 27ra ^ g 



(7.79) 



where only the lowest-order terms in (toe/a) have been kept. 

The waves propagating as exp (/k • x — icot) are damped, transverse 
waves. The fields can be written as 

E = E n e~ pn - x e ian - x - i(Ut ] 

„ . , (7.80) 

H = H e- (in -*e ma - x - l( " t J 

where n is a unit vector in the direction of k. The divergence equation for 
E shows that E • n = 0, while the relation between H and E (7.73) gives 



H = — (a + i/8)n x E 

[XOi 



(7.81) 



This shows that H and E are out of phase in a conductor. Defining the 
magnitude and phase of k : 



\k\ = Va 2 + P = V^e 



1 + (^| 



2 
CO€ / - 



"1 



<f> = tan -1 - = | tan -1 
a 



/ 47T(t \ 
\ iO€ J 



(7.82) 



[Sect. 7.8] Plane Electromagnetic Waves 225 

equation (7.81) can be written in the form: 

\ toe/ 



Ho=./ 1 



1 + 



e*n x E (7.83) 



The interpretation of (7.83) is that H lags E in time by the phase angle <f> 
and has a relative amplitude : 

\M = fih + l^fT' (7.84) 

|E | V [x L \ coe I J 

In very good conductors we see that the magnetic field is very large com- 
pared to the electric field and lags in phase by almost 45°. The field energy 
is almost entirely magnetic in nature. 

The waves given by (7.80) show an exponential damping with distance. 
This means that an electromagnetic wave entering a conductor is damped 
to 1/e = 0.369 of its initial amplitude in a distance: 

6 = - ~ . C (7.85) 

the last form being the approximation for good conductors. The distance 
6 is called the skin depth or the penetration depth. * For a conductor like 
copper, d ^ 0.85 cm for frequencies of 60 cps, and d ~ 0.71 x 10~ 3 cm 
for 100 Mc/sec. This rapid attenuation of waves means that in high- 
frequency circuits current flows only on the surface of the conductors. 
One simple consequence is that the high-frequency inductance of circuit 
elements is somewhat smaller than the low-frequency inductance because 
of the expulsion of flux from the interior of the conductors. 

The problem of reflection and refraction at an interface between con- 
ducting media is rather complicated and will not be treated here. The 
interested reader may refer to Stratton, pp. 500 ff., for a discussion of this 
point. See, however, Section 8.1 for a treatment of fields at the interface 
between a dielectric and a good conductor. 



7.8 Simple Model for Conductivity 

The simplest model of conduction, due originally to Drude (1900), is 
that in a metal there are a certain number n of electrons per unit volume 
free to move under the action of applied electric fields, but subject to 

* For reference, the skin depth (7.85) appears in mks units as d = {Ijficoa) 1 ^. 



226 Classical Electrodynamics 

damping force due to collisions. Thus the equation of motion of such an 
electron is 

d\ 

m — + mgy = eE(x, t) (7.86) 

at 

where g is the damping constant.* For rapidly oscillating fields the 
displacement of the electron is small compared to a wavelength so that 
approximately 

m — + mgy = eE Q e~ i<at (7.87) 

dt 

where E is the electric field at the average position of the electron. The 
steady-state solution for the velocity of the electron is : 

v = — — E e- imt (7.88) 

m(g — ico) 

so that the conductivity is given by 

a = "° g2 (7.89) 

m{g - ico) 

Assuming one free electron per atom, a metal such as copper (« ^ 8 x 
10 22 electrons/cm 3 , c^5 x 10 17 sec -1 ) has an empirical damping constant 
g^3x 10 13 sec -1 . This shows that for frequencies of the order of, or 
smaller than, microwave frequencies (~10 10 sec -1 ) metallic conductivities 
are essentially real (i.e., current in phase with the field) and independent of 
frequency. At higher frequencies (in the infrared and beyond), however, 
the conductivity is complex and depends markedly on frequency in a 
manner qualitatively described by the simple result (7.89). 



7.9 Transverse Waves in a Tenuous Plasma 

In certain situations, such as the ionosphere or a tenuous plasma, the 
damping of the motion of the free electrons due to collisions becomes 
negligible. Then the "conductivity" becomes purely imaginary: 

2 
^plasma ^ * — (7.90) 

mco 

* The damping constant^ is some sort of average rate of collisions involving appreci- 
able momentum transfer. Collisions occur between electrons and lattice vibrations, 
lattice imperfections, and impurities. The proper calculation of g involves quantum- 
mechanical considerations, including the effects of the Pauli exclusion principle. See 
A. H. Wilson, Theory of Metals, 2nd ed., Cambridge University Press (1953). 



[Sect. 7.9] Plane Electromagnetic Waves 227 

Quotation marks are placed on "conductivity" because there is no resistive 
loss of energy if the current and electric field are out of phase. The 
propagation of transverse electromagnetic waves in a tenuous plasma is 
governed by equation (7.76) of Section 7.7, with cr plasma (7.90) inserted for 
a:* 



k -sV-^) (7 - 91 > 

where 

coJS _ w (7 92) 

m 

is called the plasma frequency . Since the wave number can be written as 
k = nco/c, where n is the index of refraction, we see that the index of 
refraction of a plasma is given by 

For high-frequency radiation (eo > co p ) the index of refraction is real and 
the waves propagate freely. For frequencies lower than the plasma 
frequency w v , n is purely imaginary. Consequently such electromagnetic 
waves incident on a plasma will be reflected from the surface. Within the 
plasma the fields will fall off exponentially with distance from the surface. 
The penetration depth plMma is given by 

<5plasn.a = . ~ — (7.94) 

V«„ 2 — (O 2 <»p 

the last value being valid for a> < co p . On the laboratory scale, plasma 
densities are in the range n ~ 10 12 -10 16 electrons/cm 3 . This means m v ~ 
6 x 10 10 -6 x 10 12 sec -1 , so that typical penetration depths are of the order 
of 0.5 cm-5 x 10~ 3 cm for static or low-frequency fields. The expulsion 
of fields from within a plasma is a well-known effect in controlled thermo- 
nuclear processes and is exploited in attempts at confinement of hot 
plasmas (see Sections 10.5 and 10.6). 

The simple result (7.93) for the index of refraction of a plasma is 
modified by the presence of an external static magnetic induction. This 
circumstance arises not only in the laboratory, but also in the ionosphere, 
where the earth's dipole field provides the external magnetic induction. 
To illustrate the influence of the external field we consider the simple 

* Sometimes this equation is solved for co 2 as a function of k : 

to 2 ~ w/ + c 2 k 2 
Then it is called a dispersion relation for co = a)(k). 



228 Classical Electrodynamics 

problem of a tenuous electronic plasma of uniform density with a strong, 
static, uniform, magnetic induction B and transverse waves propagating 
parallel to the direction of B . If the amplitude of the electronic motion is 
small and collisions are neglected, the equation of motion is approximately : 

m — ~ eEe- imt + e - x B (7.95) 

dt c 

where the influence of the B field of the transverse wave has been neglected 
compared to the static induction B . It is convenient to consider the 
transverse waves as circularly polarized. Then 

E = £( Cl ± /eg) (7.96) 

while B is in the direction of c 3 . Since we are looking for a steady-state 
solution, we will assume that the velocity of the electron is of the form : 

v(t) = v(e 1 ±ie 2 )e- ilot (7.97) 

Then from (7.95), using (7.96), we find immediately 

v = E (7.98) 

m(co ± a) B ) 

where co B is the frequency of precession of a charged particle in a magnetic 

field, 

o> B = ^2. (7.99) 

mc 

Result (7.98) can be understood by noting that, in a coordinate system 
precessing with frequency a> B , the electron is driven by a rotating electric 
field of effective frequency co ± oo B , depending on the sign of the circular 
polarization. 

The current density in the plasma due to electronic motion is 

2 

J = en v = ^ E (7.100) 

m(co ± co B ) 

When this current density is added to the displacement current, Maxwell's 
generalization of Ampere's law becomes : 



V x H= -i- 

c 



1 T"^ |E (7.101) 



2 
p 

oo(co ± co B ). 



The factor in square brackets can be interpreted as the dielectric constant 
or square of the index of refraction : 

n ± 2 = 1 ^! (7.102) 

oo(co ± co B ) 



[Sect. 7.9] Plane Electromagnetic Waves 229 

This is the extension of (7.93) to include a static magnetic induction. It is 
not completely general, since it applies only to waves propagating along 
the static field direction. But even in this simple example we see the 
essential characteristic that waves of right-handed and left-handed circular 
polarizations propagate differently. The ionosphere is birefringent. For 
propagation in directions other than parallel to the static field B it is 
straightforward to show that, if terms of the order of co B 2 are neglected 
compared to co 2 and axo B , the index of refraction is still given by (7.102). 
But the precession frequency (7.99) is now to be interpreted as that due to 
only the component of B parallel to the direction of propagation. This 
means that co B in (7.102) is a function of angle — the medium is not only 
birefringent, but also anisotropic. 

For the ionosphere a typical maximum density of free electrons is 
n ~ 10M0 6 electrons/cm 3 , corresponding to a plasma frequency of the 
order of (o p ~ 6 x 10 6 -6 x 10 7 sec -1 . If we take a value of 0.3 gauss as 
representative of the earth's magnetic field, the precession frequency is 
co B ~6x 10 6 sec -1 . 

Figure 7. 1 1 shows n ± 2 as a function of frequency for two values of the 
ratio of (coJa) B ). In both examples there are wide intervals of frequency 
where one of n + 2 or n_ 2 is positive while the other is negative. At such 
frequencies one state of circular polarization cannot propagate in the 
plasma. Consequently a wave of that polarization incident on the plasma 
will be totally reflected. The other state of polarization will be partially 
transmitted. Thus, when a linearly polarized wave is incident on a plasma, 
the reflected wave will be elliptically polarized, with its major axis generally 
rotated away from the direction of the polarization of the incident 
wave. 

The behavior of radio waves reflected from the ionosphere is explicable 
in terms of these ideas, but the presence of several layers of plasma with 
densities and relative positions varying with height and time makes the 
problem considerably more complicated than our simple example. The 
electron densities at various heights can be inferred by studying the 
reflection of pulses of radiation transmitted vertically upwards. The 
number of free electrons per unit volume increases slowly with height in a 
given layer of the ionosphere, as shown in Fig. 7.12, reaches a maximum, 
and then falls abruptly with further increase in height. A pulse of a given 
frequency (o x enters the layer without reflection because of the slow change 
in « . When the density n is large enough, however, co p (h^) ~ co x . Then 
the indices of refraction (7.102) vanish and the pulse is reflected. The 
actual density n where the reflection occurs is given by the roots of the 
right-hand side of (7.102). By observing the time interval between the 
initial transmission and reception of the reflected signal the height h x 



230 



Classical Electrodynamics 



= 2.0 




,1 



— <N 



-1 



M 

f ' ! 


1^ = 0.5 

B 


■,^t~ \ 


i 


1 CO 

/ "J 

In 2 



-2 
-3 
-4 



Fig. 7.11 Indices of refraction as a function of frequency for model of the ionosphere 
(tenuous electronic plasma in a static, uniform magnetic induction). n±((o) apply to 
right and left circularly polarized waves propagating parallel to the magnetic field. 
o) B is the gyration frequency; eo„ is the plasma frequency. The two sets of curves 
correspond to (oJa> B = 2.0, 0.5. 

corresponding to that density can be found. By varying the frequency coj 
and studying the change in time intervals the electron density as a function 
of height can be determined. If the frequency w x is too high, the index of 
refraction does not vanish and very little reflection occurs. The frequency 
above which reflections disappear determines the maximum electron 
density in a given layer. 



m 



no(h\) 




Fig. 7.12 Electron density as a 

function of height in a layer of the 

ionosphere (schematic). 



[Probs. 7] Plane Electromagnetic Waves 231 



REFERENCES AND SUGGESTED READING 

The whole subject of optics as an electromagnetic phenomenon is treated authorita- 
tively by 

Born and Wolf. 
Their first chapter covers plane waves, polarization, and reflection and refraction, among 
other topics. A very complete discussion of plane waves incident on boundaries of 
dielectrics and conductors is given by 

Stratton, Chapter IX. 
Another good treatment of electromagnetic waves in both isotropic and anisotropic 
media is that of 

Landau and Lifshitz, Electrodynamics of Continuous Media, Chapters X and XI. 
A more elementary, but clear and thorough, approach to plane waves and their properties 
appears in 

Adler, Chu, and Fano, Chapters 7 and 8. 
The propagation of waves in dispersive media is discussed in detail in the book by 

Brillouin. 
The distortion and attenuation of pulses in dissipative materials are covered by 

Stratton, pp. 301-309. 



PROBLEMS 

7.1 An approximately monochromatic plane wave packet in one dimension has 
the instantaneous form, u(x, 0) = f(x)e ik o x , with /(*) the modulation 
envelope. For each of the forms /(*) below, calculate the wave-number 
spectrum \A(k)\ 2 of the packet, sketch \u(x, 0)| 2 and |/4(&)| 2 , evaluate explicitly 
the rms deviations from the means, Ax and Ak (defined in terms of the 
intensities \u(x, 0)| 2 and \A(k)f), and test inequality (7.28). 

(a) f(x) = Ne-*\ X U 2 

(b) f(x) = Ne-" 2 * 2 I* 
(N(l - a |a;|) for a |a;| < 1 



(c) f(x) = , 

7 (0 for a \x\ > 1 



(d)f(x) 



-K 



N for |a;| < a 

for 1*1 > a 



7.2 A plane wave is incident on a layered interface as shown in the figure (p. 232). 
The indices of refraction of the three nonpermeable media are n x , n 2 , « 3 . 
The thickness of the intermediate layer is d. 

(a) Calculate the transmission and reflection coefficients (ratios of 
transmitted and reflected Poynting's flux to the incident flux), and sketch 
their behavior as a function of frequency for n t = 1, n 2 = 2, n^ = 3 ; n x = 3, 
n 2 = 2, » 3 = 1 ; and n x = 2, » 2 = 4, n 3 = 1 . 



232 



Classical Electrodynamics 



Bi 




(Jb) The medium n x is part of an optical system (e.g., a lens); medium n 3 
is air (w 3 = 1). It is desired to put an optical coating (medium n 2 ) on the 
surface so that there is no reflected wave for a frequency <o . What thickness 
d and index of refraction /i 2 are necessary? 

7.3 Two plane semi-infinite slabs of the same uniform, isotropic, nonpermeable, 
lossless dielectric with index of refraction n are parallel and separated by an 
air gap (n = 1) of width d. A plane electromagnetic wave of" frequency to 
is incident on the gap from one of the slabs with angle of incidence i. For 
linear polarization both parallel to and perpendicular to the plane of 
incidence, 

(a) calculate the ratio of power transmitted into the second slab to the 
incident power and the ratio of reflected to incident power; 

(b) for i greater than the critical angle for total internal reflection, sketch 
the ratio of transmitted power to incident power as a function of d measured 
in units of wavelength in the gap. 

7.4 A plane polarized electromagnetic wave of frequency to in free space is 
incident normally on the flat surface of a nonpermeable medium of 
conductivity a and dielectric constant e. 

(a) Calculate the amplitude and phase of the reflected wave relative to the 
incident wave for arbitrary a and e. 

(b) Discuss the limiting cases of a very poor and a very good conductor, 
and show that for a good conductor the reflection coefficient (ratio of 
reflected to incident intensity) is approximately 



R 



c 



where 8 is the skin depth. 
7.5 A plane polarized electromagnetic wave E = E i e ik '*- i<ot is incident normally 
on a flat uniform sheet of an excellent conductor (<r > to) having a thickness t . 
Assuming that in space and in the conducting sheet /* = e = 1, discuss the 
reflection and transmission of the incident wave. 

(a) Show that the amplitudes of the reflected and transmitted waves, 
correct to the first order in (to\afA, are: 

Er -(1 - fl(l - e-™) 

E t 

Ei 



(1 - e-™) + 0(1 + 3e- 2 *) 
4/Se" A 



(1 - e~ u ) + 0(1 + 3e~ 2A ) 



[Probs. 7] Plane Electromagnetic Waves 233 

where 

A = (1 - i)t/d 

and d = c/ V27rco(r is the penetration depth. 

(ft) Verify that for zero thickness and infinite thickness you obtain the 
proper limiting results. 

(c) Show that, except for sheets of very small thickness, the transmission 
coefficient is 

_ 32(Re Pfe-W 

1 - 2e- 2 ^ cos (It IS) + *-«/« 

Sketch log 7* as a function of (t/d), assuming Re £ = lO" 2 . 

Define "very small thickness." 
7.6 Plane waves propagate in a homogeneous, nonpermeable, but anisotropic 
dielectric. The dielectric is characterized by a tensor e w , but if coordinate 
axes are chosen as the principal axes the components of displacement along 
these axes are related to the electric-field components by D t = e t E t 
(/ = 1, 2, 3), where e f are the eigenvalues of the matrix e„. 

(a) Show that plane waves with frequency a> and wave vector k must 
satisfy 

to 2 

kx(kxE)+ 1 D=0 
c £ 

(b) Show that for a given wave vector k = kn there are two distinct 
modes of propagation with different phase velocities v = co/k which satisfy 
the Fresnel equation, 



2 



2 

= 



_ „ .2 



where v t = c/V €i is called a principal velocity, and /i, is the component of 
n along the /th principal axis. 

(c) Show that D a • D 6 = 0, where D a , D 6 are the displacements associated 
with the two modes of propagation. 

7.7 A homogeneous, isotropic, nonpermeable dielectric is characterized by an 
index of refraction n(co) which is in general complex in order to describe 
absorptive processes. 

(a) Show that the general solution for plane waves in one dimension can 
be written 

1 f 00 
u(x, t) = —j= dco e- i<ot [A(co)e i ^ a,lc '> n ^ x + B((o)e~ *(<°kX<»)x\ 

where u(x, t) is a component of E or B. 

(b) If u(x, t) is real, show that n(—m) = n*((o). 

(c) Show that, if «(0, t) and 3w(0, t)\dx are the boundary values of u 
and its derivative at * = 0, the coefficients A(co) and B(a>) are 

A(co)\ 1 1 f 00 . T ic du 1 



234 Classical Electrodynamics 

7.8 A very long plane-wave train of frequency co with a sharp front edge is 
incident normally from vacuum on a semi-infinite dielectric described by 
an index of refraction n((o) and occupying the half-space x > 0. Just 
outside the dielectric (at x = 0) the incident electric field is 

£,,(0, i) — d(t)e~ et sin co t 

where 0(0 is the step function (0(0 = for t < 0, 0(0 = 1 for t > 0). The 
exponential decay constant e is a positive infinitesimal. 

(a) Using the results of Section 7.5 determine the transmitted field 
E '(x, at any point in the dielectric as an integral over real frequencies. 

(b) Prove that a sufficient condition for causality (that no signal propagate 
faster than the speed of light in vacuum) in this problem is that the index of 
refraction as a function of complex co be an analytic function, regular in the 
upper half co plane with nonvanishing imaginary part there, and approaching 
unity for \co\ -»■ <x>. 

(c) Generalize the argument of (b) to apply to any incident wave train. 

7.9 (a) Show that, if the index of refraction n{co) is analytic in the upper half 
complex co plane and approaches unity for large \co\, its real and imaginary 
parts are related for real frequencies by the dispersion relation, 



■" Jo 



Re n(a>) = 1 + - P\ —z- — 5 Im "(<>>') doi ' 



where P stands for Cauchy principal value. Write the other dispersion 
relation, expressing the imaginary part as an integral over the real. 

(b) Show by direct calculation with the dispersion relation that in a 
frequency range where resonant absorption occurs there is necessarily 
anomalous dispersion. 

(c) The elementary classical model for an index of refraction is based on a 
collection of damped electronic oscillators and gives an index of refraction, 

, , 2^e 2 y f k 

n{co) ~ 1 H > — t 3 : 

' m Z-i cojf — co* — iv k co 

k 

where co k is the resonant frequency of the fcth type of oscillator, v k its damping 
constant, and/ fc the number of such oscillators per atom. Verify that this 
index of refraction has the appropriate properties to satisfy the dispersion 
relation of (a). 



8 



Wave Guides 

and Resonant Cavities* 



Electromagnetic fields in the presence of metallic boundaries form 
a practical aspect of the subject of considerable importance. At high fre- 
quencies where the wavelengths are of the order of meters or less the only 
practical way of generating and transmitting electromagnetic radiation 
involves metallic structures with dimensions comparable to the wave- 
lengths involved. In this chapter we consider first the fields in the neigh- 
borhood of a conductor and discuss their penetration into the surface and 
the accompanying resistive losses. Then the problems of waves guided in 
hollow metal pipes and of resonant cavities are treated from a fairly 
general viewpoint, with specific illustrations included along the way. 
Finally, dielectric wave guides are briefly described as an alternative 
method of transmission. 

* In this chapter certain formulas, denoted by an asterisk on the equation number, are 
written so that they can be read as formulas in mks units provided the first factor in 
square brackets is omitted. For example, (8.12) is 



dPloaa _ T 1 l/wod . 



H„ 



The corresponding equation in mks form is 

dPiosa _ /Mid 
~da 4~ 

where all symbols are to be interpreted as mks symbols, perhaps with entirely different 
magnitudes and dimensions from those of the corresponding Gaussian symbols. 

If an asterisk appears and there is no square bracket, the formula can be interpreted 
equally in Gaussian or mks symbols. 

General rules for conversion of any equation into its corresponding mks form are 
given in Table 3 of the Appendix. 

235 



236 Classical Electrodynamics 

8.1 Fields at the Surface of and within a Conductor 

As was mentioned at the end of Section 7.7, the-problem of reflection 
and refraction of waves at an interface of two conducting media is some- 
what complicated. The most important and useful features of the 
phenomenon can, however, be obtained with an approximate treatment 
valid if one medium is a good conductor. Furthermore, the method, within 
its range of validity, is applicable to situations more general than plane 
waves incident. 

First consider a surface with unit normal n directed outward from a 
perfect conductor on one side into a nonconducting medium on the other 
side. Then, just as in the static case, there is no electric field inside the 
conductors. The charges inside a perfect conductor are assumed to be so 
mobile that they move instantly in response to changes in the fields, no 
matter how rapid, and always produce the correct surface-charge density S 
(capital 2 is used to avoid confusion with the conductivity or) : 

n • D = [4t7]2 (8.1)* 

in order to give zero electric field inside the perfect conductor. Similarly, 
for time-varying magnetic fields, the surface charges move in response to 
the tangential magnetic field to produce always the correct surface current 
K: 



n x H = 



L C J 



K (8.2)" 



in order to have zero magnetic field inside the perfect conductor. The 
other two boundary conditions are on normal B and tangential E : 



n • (B — B c ) = 0| (8.3)" 



n x (E - E c ) = 0) 

where the subscript c refers to the conductor. From these boundary 
conditions we see that just outside the surface of a perfect conductor only 
normal E and tangential H fields can exist, and that the fields drop abruptly 
to zero inside the perfect conductor. This behavior is indicated schemati- 
cally in Fig. 8.1. 

For a good, but not perfect, conductor the fields in the neighborhood 
of its surface must behave approximately the same as for a perfect con- 
ductor. In Section 7.7 we have seen that inside a conductor the fields are 
attenuated exponentially in a characteristic length d, called the skin depth. 
For good conductors and moderate frequencies, d is a small fraction of 
a centimeter. Consequently, boundary conditions (8.1) and (8.2) are 



[Sect. 8.1] 



Wave Guides and Resonant Cavities 



237 




E 
H 




{=0 i—. 

(a) (b) 

Fig. 8.1 Fields near the surface of a perfect conductor. 

approximately true for a good conductor, aside from a thin transitional layer 
at the surface. 

If we wish to examine that thin transitional region, however, care must 
be taken. First of all, Ohm's law (7.68) shows that with a finite conduct- 
ivity there cannot actually be a surface layer of current, as implied in (8.2). 
Instead, the boundary condition on the magnetic field is 



n x (H - H c ) = 



(8.4)* 



To explore the changes produced by a finite, rather than an infinite, 
conductivity we employ a successive approximation scheme. First we 
assume that just outside the conductor there exists only a normal electric 
field E ± and a tangential magnetic field H„, as for a perfect conductor. 
The values of these fields are assumed to have been obtained from the 
solution of an appropriate boundary-value problem. Then we use the 
boundary conditions and Maxwell's equations in the conductor to find the 
fields within the transition layer and small corrections to the fields outside. 
In solving Maxwell's equations within the conductor we make use of the 
fact that the spatial variation of the fields normal to the surface is much 
more rapid than the variations parallel to the surface. This means that 
we can safely neglect all derivatives with respect to coordinates parallel 
to the surface compared to the normal derivative. 

If there exists a tangential H (| outside the surface, boundary condition 
(8.4) implies the same H (| inside the surface. With the neglect of the dis- 
placement current in the conductor, the curl equations in (7.69) become 



E c ~ —V x H c 

47TO - 



IC 



H c =-— V x E c 

flOi 



(8.5) 



238 Classical Electrodynamics 

where a harmonic variation e~ imt has been assumed. If n is the unit normal 
outward from the conductor and £ is the normal coordinate inward into 
the conductor, then the gradient operator can be written 

neglecting the other derivatives when operating on the fields within the 
Conductor. With this approximation (8.5) become: 



E c ~ n x — 2 

Attg d| 

„ . ic w 3E C 

[JLOi d£ 



(8.6) 



These can be combined to yield 



^ 2 (nxH c ) + |(nxH c )~0 (8.7) 



— C-XIW + - 

and 

n • H c ~ (8.8) 



where 6 is the skin depth defined by (7.85). The second equation shows 
that inside the conductor H is parallel to the surface, consistent with our 
boundary conditions. The solution for H c is : 

H c = H M «T*"e«" (8.9) 

where H M is the tangential magnetic field outside the surface. From (8.6) 
the electric field in the conductor is approximately: 

E c ~ /^L (1 - i)(n x H M >-^ e^ l& (8.10) 

These solutions for H and E inside the conductor exhibit the properties 
discussed in Section 7.7 : (a) rapid exponential decay, (b) phase difference, 
(c) magnetic field much larger than the electric field. Furthermore, they 
show that, for a good conductor, the fields in the conductor are parallel 
to the surface* and propagate normal to it, with magnitudes which depend 
only on the tangential magnetic field H„ which exists just outside the 
surface. 

* From the continuity of the tangential component of H and the equation connecting 
E to V x H on either side of the surface, one can show that there exists in the conductor 
a small normal component of electric field, E e -n~ (/coe/47r<r)£' i , but this is of the next 
order in small quantities compared with (8.10). 



[Sect. 8.1] 



Wave Guides and Resonant Cavities 



239 



From the boundary condition on tangential E (8.3) we find that just 
outside the surface there exists a small tangential electric field given by 
(8.10), evaluated at | = 0: 



Ei, 



(1 - i)(n x H„) 



(8.11) 



In this approximation there is also a small normal component of B just 
outside the surface. This can be obtained from Faraday's law of induction 
and gives B ± of the same order of magnitude as E N . The amplitudes of 
the fields both inside and outside the conductor are indicated schematically 
in Fig. 8.2. 

The existence of a small tangential component of E outside the surface, 
in addition to the normal E and tangential H, means that there is a power 
flow into the conductor. The time-average power absorbed per unit area 



is 



dPu 



da 



= - °- Re [n • E x H*] = \f\ ^ |H„| 2 (8.12)< 

8tt UttJ 4 



This result can be given a simple interpretation as ohmic losses in the body 



E, 
H 




Fig. 8.2 Fields near the surface of a good, but not perfect, conductor. 



240 Classical Electrodynamics 

of the conductor. According to Ohm's law, there exists a current density 
J near the surface of the conductor: 

J = oE e = /^ (1 - 0(n x H M >-^-*>/* (8.13) 

The time-average rate of dissipation of energy per unit volume in ohmic 
losses is £J • E* = (l/2<r) | J| 2 , so that the total rate of energy dissipation 
in the conductor for the volume lying beneath an area element AA is 

± AA f°°^J . J* = AA^- |HJ 2 f V«" d£ = AA^ IHJ 2 
2a Jo 877 Jo 16tt 

This is the same rate of energy dissipation as given by the Poynting's 
vector result (8.12). 

The current density J is confined to such a small thickness just below 
the surface of the conductor that it is equivalent to an effective surface 
current K eff : 



-i 



Keff = J d£ = 



c 
Att- 



n x H„ (8.14)' 



Comparison with (8.2) shows that a good conductor behaves effectively 
like a perfect conductor, with the idealized surface current replaced by an 
equivalent surface current which is actually distributed throughout a very 
small, but finite, thickness at the surface. The power loss can be written in 
terms of the effective surface current : 



dPioss 1 ,^ r .2 



da 2ad 



Keftr (8.15)* 



This shows that 1 /ad plays the role of a surface resistance of the con- 
ductor. Equation (8.15), with K eff given by (8.14), or (8.12) will allow us 
to calculate approximately the resistive losses for practical cavities, trans- 
mission lines, and wave guides, provided we have solved for the fields in 
the idealized problem of infinite conductivity. 



8.2 Cylindrical Cavities and Wave Guides 

A practical situation of great importance is the propagation or excitation 
of electromagnetic waves in hollow metallic cylinders. If the cylinder has 
end surfaces, it is called a cavity; otherwise, a wave guide. In our 
discussion of this problem the boundary surfaces will be assumed to be 
perfect conductors. The losses occurring in practice can be accounted for 



Wave Guides and Resonant Cavities 



241 



[Sect. 8.2] 

adequately by the methods of Section 8.1. A cylindrical surface S of 
general cross-sectional contour is shown in Fig. 8.3. For simplicity, the 
cross-sectional size and shape are assumed constant along the cylinder axis. 
With a sinusoidal time dependence e~ i(0t for the fields inside the cylinder, 
Maxwell's equations take the form: 



V x E= i-B 



V-B = 



VxB=-i>e-E V-E = 



(8.16) 



where it is assumed that the cylinder is filled with a uniform nondissipative 
medium having dielectric constant e and permeability fx. If follows that 
both E and B satisfy 



("■"30- 



(8.17) 



Because of the cylindrical geometry it is useful to single out the spatial 
variation of the fields in the z direction and to assume 



E(z, y, z, t) 
B(x, y, z, t) 



E(x, y)e ±ilez ' i<ot 
B(x, y) e ±ikz - i(0t 



(8.18) 



Appropriate linear combinations can be formed to give traveling or 
standing waves in the z direction. The wave number k is, at present, an 
unknown parameter which may be real or complex. With this assumed z 
dependence of the fields the wave equation reduces to the two-dimensional 
form: 



[*.' + ("?-*)]0- 



where V, 2 is the transverse part of the Laplacian operator: 



V, 2 = V 2 - — 

* dz 2 



(8.19) 



(8.20) 




Fig. 8.3 Hollow, cylindrical wave guide of arbitrary cross-sectional shape. 



242 



Classical Electrodynamics 



It is also useful to separate the fields into components parallel to and 
transverse to the z axis : 

t , E = E, + E, (8.21) 

where the parallel field is 

E z = (e 3 • E)e 3 (8.22) 

and the transverse field is 



E t = (e 3 x E) x e 3 



(8.23) 



and e 3 is a unit vector in the z direction. Similar definitions hold for the 
magnetic-flux density B. Manipulation of the curl equations in (8.16) and 
use of the explicit z dependence (8.18) lead to the determination of the 
transverse fields in terms of the axial components : 



B, = 



E, = 



("7-*) 

1 



+ i>€ - e 3 x V t E z 

c 



Wf) 



(8.24) 



These relations show that it is sufficient to determine E z and B, as the 
appropriate solutions of the two-dimensional wave equation (8.19). The 
other components can then be calculated from (8.24). 

The boundary values on the surface of the cylinder will be taken as those 
for a perfect conductor: 



n • B = 0, nxE = 



(8.25) 



where n is a unit normal at the surface. Since Maxwell's equations and 
the boundary conditions are internally consistent, it is sufficient to note 
that the vanishing of tangential E at the surface requires 

Ez \s = (8.26) 

For the normal components of B, using the expression for B t (8.24), we 
find that n • B = implies 

dB„ 



dn 



= 



(8.27) 



where d/dn is the normal derivative at a point on the surface. 

The two-dimensional wave equations (8.19) for E z and B z , together with 
the boundary conditions on E z and B z at the surface of the cylinder, 
specify eigenvalue problems of the usual sort. For a given frequency co, 
only certain values of the axial wave number k will be consistent with 



[Sect. 8.2] Wave Guides and Resonant Cavities 243 

the differential equation and the boundary conditions (typical wave-guide 
situation); or, for a given k, only certain frequencies m will be allowed 
(typical resonant-cavity situation). Because the boundary conditions on 
E z and B z are different, they cannot generally be satisfied simultaneously. 
Consequently the fields divide themselves into two distinct categories: 

TRANSVERSE MAGNETIC (TM) 

B z = everywhere 

The boundary condition is 

E z \s = 

TRANSVERSE ELECTRIC (TE) 

E z = everywhere 
The boundary condition is 

an 

= 



dB, 

dn 



s 



The designations "Electric (or E) Waves" and "Magnetic (or H) Waves" 
are sometimes used instead of Transverse Magnetic and Transverse 
Electric, respectively, corresponding to specification of the axial com- 
ponent of the field. In addition to these two types of fields there is a 
degenerate mode, called the Transverse Electromagnetic (TEM) mode, in 
which both E z and B z vanish. From (8.24) we see that, in order to have 
nonvanishing transverse components when both E z and B z vanish, the 
axial wave number must satisfy the condition: 



k = V^- (8.28) 



Thus TEM waves travel as if they were in an infinite medium without 
boundary surfaces. From the two-dimensional wave equation (8.19) we 
now find 

V t 2 ( Etem \ = (8.29) 

showing that each component of the transverse fields satisfies Laplace's 
equation of electrostatics in two dimensions. It is easy to show (a) that 
both E TEM and B TEM are derivable from scalar potentials satisfying 
Laplace's equation and (b) that B TEM is everywhere perpendicular to E TEM . 
In fact, from Faraday's law of induction we find 

B TEM = — f-(e3*E TEM ) (8-30) 

ICO oz 



244 Classical Electrodynamics 

With z-dependence e iVft€(OZlc , we have 

B T EM = V-" 6 e 3 X E TEM (8.31)* 

which is just the relation for plane waves in an infinite medium. 

An immediate consequence of (8.29) is that the TEM mode cannot 
exist inside a single hollow, cylindrical conductor of infinite conductivity. 
The surface is an equipotential; hence the electric field vanishes inside. 
It is necessary to have two or more cylindrical surfaces in order to support 
the TEM mode. The familiar coaxial cable and the parallel-wire trans- 
mission line are structures for which this is the dominant mode. (See 
Problems 8.1 and 8.2.) 

8.3 Wave Guides 

We now consider the propagation of electromagnetic waves along a 
hollow wave guide of uniform cross section. With the z-dependence e ikz , 
the transverse components of the fields for the two types of waves are 
related, according to (8.24), as follows: 



TM waves: b* = ^ e 3 x E, 

ck 

TE waves: E(=--e 3 xB ( 

ck 



(8.32) 



The transverse fields are in turn determined by the longitudinal fields: 
TM waves: Et= l -^V t y> 

., (8.33) 

TE waves: B, = — V t y> 

y 2 

where ip is E z (B z ) for TM (TE) waves. The scalar function y> satisfies the 
two-dimensional wave equation (8.19): 

(V, 2 + y 2 )y = (8.34) 

where 



CO 2 



y 2 = /* — - k 2 (8.35) 

subject to the boundary condition : 

dip 



for TM (TE) waves. 



y> \ s = 0, or 

on 



= (8.36) 



[Sect. 8.3] 



Wave Guides and Resonant Cavities 



245 



Equation (8.34) for y>, together with boundary condition (8.36), specifies 
an eigenvalue problem. It is easy to see that the constant y 2 must be non- 
negative. Roughly speaking, it is because y must be oscillatory in order 
to satisfy boundary condition (8.36) on opposite sides of the cylinder. 
There will be a spectrum of eigenvalues y x 2 and corresponding solutions 
y x , A = 1, 2, 3, . . . , which form an orthogonal set. These different 
solutions are called the modes of the guide. For a given frequency a>, the 
wave number k is determined for each value of A: 



,2 w 2 



(8.37) 



If we define a cutoff frequency a> x , 



7i 






then the wave number can be written: 



k 2 = 



\/ fX€\J CO 2 — Oi x 



(8.38)< 



(8.39)< 



We note that, for a> > co x , the wave number k x is real; waves of the A 
mode can propagate in the guide. For frequencies less than the cutoff 
frequency, k x is imaginary; such modes cannot propagate and are called 
cutoff modes. The behavior of the axial wave number as a function of 
frequency is shown qualitatively in Fig. 8.4. We see that at any given 
frequency only a finite number of modes can propagate. It is often con- 
venient to choose the dimensions of the guide so that at the operating 
frequency only the lowest mode can occur. This is shown by the vertical 
arrow on the figure. 

Since the wave number k x is always less than the free-space value 
VJuco/c, the wavelength in the guide is always greater than the free-space 



Fig. 8.4 Wave number k x versus 

frequency o> for various modes A. 

co x is the cutoff frequency. 




0>4 «5 



246 



Classical Electrodynamics 



wavelength. In turn, the phase velocity v v is larger than the infinite space 
value : 



_ o> _ _c 1 c 



The phase velocity becomes infinite exactly at cutoff. 



(8.40) 



8.4 Modes in a Rectangular Wave Guide 

As an important illustration of the general features described in 
Section 8.3 we consider the propagation of TE waves in a rectangular wave 
guide with inner dimensions a, h, as shown in Fig. 8.5. The wave equation 
for y) = B z is 



/ a 2 a 2 ,\ 



(8.41) 



with boundary conditions dxpjdn = at x = 0, a and y = 0, b. The 
solution for ip is consequently 

Vmn(*» V) = B cos iTm J cos (UI^j ( 8 42) 

where 



\ a 2 bV 



(8.43) 



The single index X specifying the modes previously is now replaced by the 
two positive integers m, n. In order that there be nontrivial solutions, m 
and n cannot both be zero. The cutoff frequency co mn is given by 



co„ 



= [c~\^—(— 4- — V 
Jue\a 2 b 2 / 



(8.44)" 




Fig. 8.5 



[Sect. 8.4] 



Wave Guides and Resonant Cavities 



247 



If a > b, the lowest cutoff frequency, that of the dominant TE mode, 
occurs for m = 1, n = 0: 

c 10 = -p- (8.45) 

This corresponds to one-half of a free-space wavelength across the guide. 
The explicit fields for this mode, denoted by TE 10 , are: 



B„ 



= B n cos\—)e ikz - 



ika 



B = B n sin 



1 7TX \ ikz-icot 



(8.46) 



ttc \ a I 



The presence of a factor i in B x (and E y ) means that there is a spatial (or 
temporal) phase difference of 90° between B x (and E y ) and B, in the 
propagation direction. It happens that the TE X mode has the lowest 
cutoff frequency of both TE and TM modes,* and so is the one used in 
most practical situations. For a typical choice a = 2b the ratio of cutoff 
frequencies co mn for the next few modes to a> 10 are as follows: 







n 


-► 









1 


2 


3 







2.00 


4.00 


6.00 


1 


1.00 


2.24 


4.13 




m 2 


2.00 


2.84 


4.48 




I 3 


3.00 


3.61 


5.00 




4 


4.00 


4.48 


5.66 




5 


5.00 


5.39 






6 


6.00 









There is a frequency range from cutoff to twice cutoff where the TB X 
mode is the only propagating mode. Beyond that frequency other modes 
rapidly begin to enter. The field configurations of the TEj mode and 
other modes are shown in many books, e.g., American Institute of Physics 
Handbook, McGraw-Hill, New York (1957), p. 5-61. 

* This is evident if we note that for the TM modes E z is of the form 

~) sin \t) 

while y 2 is still given by (8.43). The lowest mode has m = n = 1. Its cutoff frequency is 
greater than that of the TE 1>0 mode by the factor 11+ — 1 . 



248 



Classical Electrodynamics 



8.5 Energy Flow and Attenuation in Wave Guides 

The general discussion of Section 8.3 for a cylindrical wave guide of 
arbitrary cross-sectional shape can be extended to include the flow of 
energy along the guide and the attenuation of the waves due to losses in the 
walls having finite conductivity. The treatment will be restricted to one 
mode at a time; degenerate modes will be mentioned only briefly. The 
flow of energy is described by the complex Poynting's vector: 



S = 



c 
Air. 



-(Ex H*) 



(8.47)* 



whose real part gives the time-averaged flux of energy. For the two types 
of field we find, using (8.24): 



S = 



cok 


e 


L k J 


tory* 


1 


v 2 
e 3 IV^I 2 - i y — y*V t xp 
k J 



(8.48) 



where the upper (lower) line is for TM (TE) modes. Since y> is generally 
real,* we see that the transverse component of S represents reactive energy 
flow and does not contribute to the time-average flux of energy. On the 
other hand, the axial component of S gives the time-averaged flow of 
energy along the guide. To evaluate the total power flow P we integrate 
the axial component of S over the cross-sectional area A : 



-Js. 

Ja 



e, da = 




■W)* m Wti)da 



(8.49) 



By means of Green's first identity (1.34) applied to two dimensions, (8.49) 
can be written: 



P = 




i>y>*^dl- \y>*V 2 yda 
LJc dn Ja 



(8.50) 



where the first integral is around the curve C which defines the boundary 
surface of the cylinder. This integral vanishes for both types of fields 

* It is possible to excite a guide in such a manner that a given mode or linear combina- 
tion of modes has a complex xp. Then a time-averaged transverse energy flow can occur. 
Since it is a circulatory flow, however, it really only represents stored energy and is not 
of great practical importance. 



[Sect. 8.5] Wave Guides and Resonant Cavities 249 

because of boundary conditions (8.36). By means of the wave equation 
(8.34) the second integral may be reduced to the normalization integral for 
ip. Consequently the transmitted power is 



P = 



£M^- m ML^° ^ 



where the upper (lower) line is for TM (TE) modes, and we have exhibited 
all the frequency dependence explicitly. 

It is straightforward to calculate the field energy per unit length of the 
guide in the same way as the power flow. The result is 



u =[t] l 2^ML^ da (8 - 52) * 



Comparison with the power flow P shows that P and U are proportional. 
The constant of proportionality has the dimensions of velocity (velocity 
of energy flow) and is just the group velocity : 



£_*£!-' ^/l-Si.-,. (8.53) 

U co jue ^Jfic 

as can be verified by a direct calculation of v g = dco/dk from (8.39), 
assuming that the dielectric filling the guide is nondispersive. We note 
that v g is always less than the velocity of waves in an infinite medium and 
falls to zero at cutoff. The product of phase velocity (8.40) and group 
velocity is constant: 

v p v g = - (8.54) 

an immediate consequence of the fact that co Aco oc k AA:. 

Our considerations so far have applied to wave guides with perfectly 
conducting walls. The axial wave number k x was either real or purely 
imaginary. If the walls have a finite conductivity, there will be ohmic 
losses and the power flow along the guide will be attenuated. For walls 
with large conductivity the wave number will have a small imaginary part: 

fc.-fcf + i^ (8.55)* 

where k x 0) is the value for perfectly conducting walls. The attenuation 
constant /3 A can be found either by solving the boundary-value problem 
over again with boundary conditions appropriate for finite conductivity, 
or by calculating the ohmic losses by the methods of Section 8.1 and 



250 



Classical Electrodynamics 



using conservation of energy. We will use the latter technique. The 
power flow along the guide will be given by 



P(z) = P e~ 2 ^ z 
Thus the attenuation constant is given by 

1 dP 



&=- 



IP dz 



(8.56)" 



(8.57)* 



where —dP/dz is the power dissipated in ohmic losses per unit length of the 
guide. According to the results of Section 8.1, this power loss is 



dP 

dz 



1 



I67T J2adjbf Jc 



In x Bl 2 dl 



(8.58)* 



where the integral is around the boundary of the guide. With fields (8.32) 
and (8.33) it is easy to show that for a given mode: 



dP 



dz ZlTpad/jL 



'\coJ Jc 



I jU€CO x 



dtp 
dn 



--^) |n xV ( 
co 2 / 



vl 2 + ^rM 2 



Ydl 



(8.59) 
where again the upper (lower) line applies to TM (TE) modes. 

Since the transverse derivatives of xp are determined entirely by the size 
and shape of the wave guide, the frequency dependence of the power loss 
is explicitly exhibited in (8.59). In fact, the integrals in (8.59) may be simply 
estimated from the fact that for each mode : 



V, 2 + ^Hy, = 



(8.60) 



This means that, in some average sense, and barring exceptional circum- 
stances, the transverse derivatives of ip must be of the order of magnitude 
of Vjbt€(a)Jc)ip : 



dip 
dn 



CO 



(\nxVM 2 )-^^vW) 



(8.61) 



Consequently the line integrals in (8.59) can be related to the normalization 
integral of |-y;| 2 over the area. For example, 



cco x ~ 



dtp 
dn 



AJa 



dl = l^e— I \ip\ 2 da 
AJa 



(8.62) 



where C is the circumference and A is the area of cross section, while g x is 
a dimensionless number of the order of unity. Without further knowledge 



[Sect. 8.5] 



Wave Guides and Resonant Cavities 



251 



of the shape of the guide we can obtain the order of magnitude of the 
attenuation constant fi x and exhibit completely its frequency dependence. 
Thus, using (8.59) with (8.62) and (8.51), plus the frequency dependence of 
the skin depth (7.85), we find 



fti- 



c 

.4-77. 



.±(<L\ 



ft od x \2A/ 



\ or 1 



£i + Vx 



Oil - 



(8.63)" 



where a is the conductivity (assumed independent of frequency), d x is the 
skin depth at the cutoff frequency, and £ A , r\ x are dimensionless numbers 
of the order of unity. For TM modes, r\ x = 0. 

For a given cross-sectional geometry it is a straightforward matter to 
calculate the dimensionless parameters £ A and r\ x in (8.63). For the TE 
modes with n = in a rectangular guide, the values are | w0 = a/ (a + b) 
and r} mfi = 2b\{a + b). For reasonable relative dimensions, these 
parameters are of order unity, as expected. 











t 

0A 


_ 


\ 


TM^- 


— 




— J£— — 






1 

IjH 1 


1 1 



»/«> 



Fig. 8.6 Attenuation constant $\ 
as a function of frequency for 
typical TE and TM modes. For 
TM modes the minimum atten- 
uation occurs at co/cox = "^3, re- 
gardless of cross-sectional shape. 



The general behavior of /S A as a function of frequency is shown in 
Fig. 8.6. Minimum attenuation occurs at a frequency well above cutoff. 
For TE modes the relative magnitudes of t- x and rj x depend on the shape 
of the guide and on X. Consequently no general statement can be made 
about the exact frequency for minimum attenuation. But for TM modes 
the minimum always occurs at co min = V3oj x . At high frequencies the 
attenuation increases as co 1 ^. In the microwave region typical attenuation 
constants for copper guides are of the order /9 A ~ lOr^coJc, giving \fe 
distances of 200-400 meters. 

The approximations employed in obtaining (8.63) break down close to 
cutoff. Evidence for this is the physically impossible, infinite value of 
(8.63) at a> = co x . A treatment of the problem by perturbation theory 



252 Classical Electrodynamics 

with the boundary condition (8.11) yields the more accurate result, 

k * = k <m + 2(1 + i)k (o)^ (8>64) 

where & is still given by (8.63) For k[ 0) > X this reduces to our previous 
expression (8.55). But at cutoff (k{ 0) = 0) the wave number is now finite 
with real and imaginary parts of the order of the geometrical mean of 
ojJc and a typical value of fj x , say at co ~ 2co A . 

In the discussion so far we have considered only one mode at a time. 
This procedure fails whenever a TE and a TM mode have the same cutoff 
frequency, as occurs in the rectangular guide, for example, with n ^ 0, 
m ^ 0. The reason for the failure is that the boundary condition (8.11) 
for finite conductivity couples the degenerate modes. The calculation of 
the attenuation then involves so-called degenerate state perturbation 
theory, and the expression for £ takes the form, 

P = KAtm + Ate) ± A/(£ TM - £ TE ) 2 + |tf|* (8.65) 

where /9 TM and /? TB are the values found above, while K is a coupling 
parameter. The two values of ft in (8.65) give the attenuation for the 
two orthogonal, mixed modes which satisfy the perturbed boundary 
conditions. * 
8.6 Resonant Cavities 

Although an electromagnetic cavity resonator can be of any shape 
whatsoever, an important class of cavities is produced by placing end 
faces on a length of cylindrical wave guide. We will assume that the end 
surfaces are plane and perpendicular to the axis of the cylinder. As usual, 
the walls of the cavity are taken to have infinite conductivity, while the 
cavity is filled with a lossless dielectric with constants ju, e. Because of 
reflections at the end surfaces the z dependence of the fields will be that 
appropriate to standing waves : 

A sin kz + B cos kz 

If the plane boundary surfaces are at z = and z = d, the boundary 
conditions can be satisfied at each surface only if 

k = Pl> P = 0,1,2,... (8.66) 

* For the theory of perturbation of boundary conditions in guides and cavities, see 
G. Goubau, Electromagnetic Waveguides and Cavities, Pergamon Press, New York, 
1961; Sect. 25. Attenuation for degenerate modes in guides is treated by R. Muller, 
Z. Naturforsch., 4a, 218 (1949), and for the rectangular cavity by the same author in 
Sect. 37 of the book by Goubau. 



[Sect. 8.6] Wave Guides and Resonant Cavities 253 

For TM fields the vanishing of E t at z = and z = d requires 

E z = y>{x, y) cos (^), p = 0, 1, 2, . . . (8.67) 

Similarly for TE fields, the vanishing of B z at z = and z = d requires 

B z = y>(x, y) sin (^), p = 1, 2, 3, . . . (8.68) 

Then from (8.24) we find the transverse fields : 
TM FIELDS 






(8.69) 



TE FIELDS 



E^-^sin^^xV^ 

B( = £Lcos(^W 
' dy 2 \ d 1 



(8.70) 



The boundary conditions at the ends of the cavity are now explicitly 
satisfied. There remains the eigenvalue problem (8.34)-(8.36), as before. 
But now the constant y 2 is : 



/ 



= "<7-(t) 2 (8 - 71) 



For each value of p the eigenvalue y x 2 determines an eigenfrequency of 
resonance frequency a) Xp : 

d 



l v = ^ J \y, 2 +( E jJ] (»-72)« 






and the corresponding fields of that resonant mode. The resonance 
frequencies form a discrete set which can be determined graphically on the 
figure of axial wave number k versus frequency in a wave guide (see p. 245) 
by demanding that k = p-rr/d. It is usually expedient to choose the 
various dimensions of the cavity so that the resonant frequency of operation 
lies well separated from other resonant frequencies. Then the cavity will 
be relatively stable in operation and insensitive to perturbing effects 
associated with frequency drifts, changes in loading, etc. 



254 



Classical Electrodynamics 




Fig. 8.7 



An important practical resonant cavity is the right circular cylinder, 
perhaps with a piston to allow tuning by varying the height. The cylinder 
is shown in Fig. 8.7, with inner radius R and length d. For a TM mode 
the transverse wave equation for y> = E z , subject to the boundary con- 
dition E g = at p = R, has the solution : 



where 



V(P, <f>) = J m (ymnP)e ±im * 



Ymn 



R 



(8.73) 
(8.74) 



x mn is the «th root of the equation, J m (x) = 0. These roots are given on 
page 72, below equation (3.92). The integers m and n take on the values 
m — 0, 1, 2, . . . , and n = 1, 2, 3, . . . . The resonance frequencies are 
given by 



^-' tmin/n ^^ 



V/" € 



X mn , P iP 

R 2 d 2 



(8.75)* 



The lowest TM mode has m = 0, n = 1, p = 0, and so is designated 
TM 0|10 . Its resonance frequency is 

_ 2.405 c 

^010 



COnin — 



The explicit expressions for the fields are 



(8.76) 






(8.77) 



The resonant frequency for this mode is independent of d. Consequently 
simple tuning is impossible. 



[Sect. 8.7] Wave Guides and Resonant Cavities 255 

For TE modes, the basic solution (8.73) still applies, but the boundary 



condition on B, 



«.-•> 



makes 

(8.78) 



R 



where x' mn is the «th root of J m '(x) = 0. These roots, for a few values of 
m and n, are tabulated below : 



Roots of J m '(x) = 



m = 

m = 1 

m = 2 

m = 3 



x' 0n = 3.832, 7.016, 10.174, 
x' ln = 1.841, 5.331, 8.536, 
x' 2n = 3.054, 6.706, 9.970, 
x' Zn = 4.201, 8.015, 11.336, 

The resonance frequencies are given by 



„„..Ii/^. + £^ (8 . 7 9). 



tymv 



where m = 0, 1, 2, ... , but n,p = 1, 2, 3, ... . The lowest TE mode has 
m = n = p = I, and is denoted TE X x x . Its resonance frequency is 



_ 1.841 c( 



V^M 1 + 2 ' 912 ^ (880) 

while the fields are derivable from 

B 2 = Vi^i^f) cos ^ sin {^)e- i(0t (8.81) 

by means of (8.70). For d large enough (d > 2.03/J), the resonance 
frequency eo m is smaller than that for the lowest TM mode (8.76). Then 
the TE ljlfl mode is the fundamental oscillation of the cavity. Because the 
frequency depends on the ratio d/R it is possible to provide easy tuning by 
making the separation of the end faces adjustable. 



8.7 Power Losses in a Cavity; Q of a Cavity 

In the preceding section it was found that resonant cavities had discrete 
frequencies of oscillation with a definite field configuration for each 
resonance frequency. This implies that, if one were attempting to excite a 
particular mode of oscillation in a cavity by some means, no fields of the 



256 Classical Electrodynamics 

right sort could be built up unless the exciting frequency were exactly equal 
to the chosen resonance frequency. In actual fact there will not be a delta 
function singularity, but rather a narrow band of frequencies around the 
eigenfrequency over which appreciable excitation can occur. An important 
source of this smearing out of the sharp frequency of oscillation is the 
dissipation of energy in the cavity walls and perhaps in the dielectric filling 
the cavity. A measure of the sharpness of response of the cavity to external 
excitation is the Q of the cavity, defined as 2tt times the ratio of the 
time-averaged energy stored in the cavity to the energy loss per cycle: 

Q _ ^ Stored energy 
Power loss 

Here co is the resonance frequency, assuming no losses. By conservation 
of energy the power dissipated in ohmic losses is the negative of the time 
rate of change of stored energy U. Thus from (8.82) we can write an 
equation for the behavior of U as a function of time: 



dU _ cop T j 

dt ~ Q 
with solution 

U(t) = Uoe-^^ 



(8.83) 



If an initial amount of energy U is stored in the cavity, it decays away 
exponentially with a decay constant inversely proportional to Q. The 
time dependence in (8.83) implies that the oscillations of the fields in the 
cavity are damped as follows : 

£(0 = Eoe -o> t/2Q e -i«>ot (8.84) 

A damped oscillation such as this has not a pure frequency, but a super- 
position of frequencies around co — co . Thus, 

1 f 00 
£(0 = -±= E(co)e- 1(0t dco 

JllT J -oo 



4 

where 



1 f 00 

E(C0) = -= E -<oot/2Q e Ha>-<oo)t df 

yjllT JO 



(8.85) 



The integral in (8.85) is elementary and leads to a frequency distribution 
for the energy in the cavity having a Lorentz line shape : 

|£(w)|2 K ( v!w /?m« (8 - 86) 

(co - w Q y + (co /2Q) 



[Sect. 8.7] 



Wave Guides and Resonant Cavities 



257 



The resonance shape (8.86), shown in Fig. 8.8, has a full width at half- 
maximum (confusingly called the half- width) equal to coJQ. For a 
constant input voltage, the energy of oscillation in the cavity as a function 
of frequency will follow the resonance curve in the neighborhood of a 
particular resonant frequency. Thus, if Aco is the frequency separation 
between half-power points, the Q of the cavity is 



2 = ? 

Aco 



(8.87) 



Q values of several hundreds or thousands are common for microwave 
cavities. 

To determine the Q of a cavity we must calculate the time-averaged 
energy stored in it and then determine the power loss in the walls. The 
computations are very similiar to those done in Section 8.5 for attenuation 
in wave guides. We will consider here only the cylindrical cavities of 
Section 8.6, assuming no degeneracies (see the footnote on p. 252). The 
energy stored in the cavity for the mode A, p is, according to (8.67)-(8.70) : 

where the upper (lower) line applies to TM (TE) modes. For the TM 
modes with p = the result must be multiplied by 2. 
The power loss can be calculated by a modification of (8.58): 



p >~=[£\Mi d i 



dz\n x Blfides + 2 da\n x B|| nd8 



> j da\n x B|| nd8 ] 



(8.89)" 



For TM modes with p ^ it is easy to show that 



Fig. 8.8 Resonance line shape. The 
full width Aco at half-maximum (of 
the power) is equal to the central 
frequency eo divided by the Q of the 
cavity. 




coJQ = Aco 



258 Classical Electrodynamics 

where the dimensionless number $ x is the same one that appears in (8.62), 
C is the circumference of the cavity, and A is its cross-sectional area. For 
p = 0, £ A must be replaced by 2l- x . Combining (8.88) and (8.89) according 
to (8.82), and using definition (7.85) for the skin depth d, we find the Q of 
the cavity : 

_£d 1 



Q= *-/ 77: (8 - 91) " 



AA> 

where fx c is the permeability of the metal walls of the cavity. For p = 
modes, (8.91) must be multiplied by 2 and £ x replaced by 21- x . This 
expression for Q has an intuitive physical interpretation when written in 
the form : 

Q = ^(— ) x (Geometrical factor) (8.92)* 



where Fis the volume of the cavity, and S its total surface area. The Q of 
a cavity is evidently, apart from a geometrical factor, the ratio of the 
volume occupied by the fields to the volume of the conductor into which 
the fields penetrate because of the finite conductivity. For TM modes in 
cylindrical cavities the geometrical factor is 






(8.93) 



for p =£ 0, and is 



211 +mL 
' + *§) 



AA) 
Cd\ 

(8.94) 



for/? = modes. For TE modes in the cylindrical cavity the geometrical 
factor is somewhat more complicated, but of the same order of magnitude. 
For the TM 01>0 mode in a circular cylindrical cavity with fields (8.77), 
| A = 1 (true for all TM modes), so that the geometrical factor is 2 and 
Q is: 



(■♦£) 



[Sect. 8.8] Wave Guides and Resonant Cavities 259 

For the TE lfl>1 mode calculation yields a geometrical factor* 



K) 



(l + 0.209 - + 0.242 4 1 
V R R 3 / 



(8.96) 



and a Q : 



( i+o - 344 S) 

^ ^ (1 + 0.209 - + 0.242 — ) 
\ R RV 



e-^-iTb — : ; — "' „x ( 8 - 97 )' 



Expression (8.92) for g applies not only to cylindrical cavities but also 
to cavities of arbitrary shape, with an appropriate geometrical factor of 
the order of unity. 

8.8 Dielectric Wave Guides 

In Sections 8.2-8.5 we considered wave guides made of hollow metal 
cylinders with fields only inside the hollow. Other guiding structures are 
possible. The parallel-wire transmission line is an example. The general 
requirement for a guide of electromagnetic waves is that there be a flow of 
energy only along the guiding structure and not perpendicular to it. This 
means that the fields will be appreciable only in the immediate neighbor- 
hood of the guiding structure. For hollow wave guides these requirements 
are satisfied in a trivial way. But for an open structure like the parallel- 
wire line the fields extend somewhat away from the conductors, falling off 
like p~ 2 for the TEM mode, and exponentially for higher modes. 

A dielectric cylinder, such as shown in Fig. 8.9, can serve as a wave guide, 
with some properties very similar to those of a hollow metal guide if its 
dielectric constant is large enough. There are, however, characteristic 
differences which arise because of the very different boundary conditions 
to be satisfied at the surface of the cylinder. The general considerations of 
Section 8.2 still apply, except that the transverse behavior of the fields is 
governed by two equations like (8.19), one for inside the cylinder and one 
for outside : 

INSIDE 

V *'+(^ e i£ -**)]{§ = ° < 8 - 98 > 

* Note that this factor varies by only 30 per cent as the cylinder geometry is changed 
from djR^>\ to d/R < 1. 



260 



Classical Electrodynamics 



OUTSIDE 



V+\wo^-k* 



= 



(8.99) 



Both dielectric (/f l5 ej) and surrounding medium (/u , e ) are assumed to be 
uniform and isotropic in their properties. The axial propagation constant 
k must be the same inside and outside the cylinder in order to satisfy 
boundary conditions at all points on the surface at all times. 

In the usual way, inside the dielectric cylinder the transverse Laplacian 
of the fields must be negative so that the constant 



V = /"i € i 



-k 2 



(8.100) 



is positive. Outside the cylinder, however, the requirement of no transverse 
flow of energy demands that the fields fall off exponentially. (There is no 
TEM mode for a dielectric guide.) Consequently, the quantity in (8.99) 
equivalent to y 2 must be negative. Therefore we define a quantity /? 2 : 



£ 2 = k 2 - fi e — 
c l 



(8.101) 



and demand that acceptable wave guide solutions have /3 2 positive (/? real). 
The oscillatory solutions (inside) must be matched to the exponential 
solutions (outside) at the boundary of the dielectric cylinder. The 
boundary conditions are continuity of normal B and D and tangential E 
and H, rather than the vanishing of normal B and tangential E (8.25) 
appropriate for hollow conductors. Because of the more involved 
boundary conditions the types of fields do not separate into TE and TM 
modes, except in special circumstances such as azimuthal symmetry in 




Moeo 



Fig. 8.9 Section of dielectric wave 
guide. 



[Sect. 8.8] 



Wave Guides and Resonant Cavities 



261 



circular cylinders, to be discussed below. In general, axial components 
of both E and B exist. Such waves are sometimes designated as HE 
modes. 

To illustrate some of the features of the dielectric wave guide we consider 
a circular cylinder of radius a consisting of nonpermeable dielectric with 
dielectric constant e x in an external nonpermeable medium with dielectric 
constant e . As a simplifying assumption we take the fields to have no 
azimuthal variation. Then in cylindrical coordinates the radial equations 
for E z or B z are Bessel's equations : 



d 2 , 1 d , o . _ ^ 

dp p dp 

jL + iA -0*^ = 0, P >a 
dp p dp 



(8.102) 



The solution, satisfying the requirements of finiteness at the origin and at 
infinity, is found from Section 3.6 to be: 



V = "j 



I Jo(yp)> 



AKtfp), 




(8.103) 



The other components of E and B can be found from (8.24) when the 
relative amounts of E z and B z are known. With no <f> dependence to the 
fields, (8.24) reduces to 



INSIDE 



ik dB, 


D ie^ dE z 
y l c dp 


E * = ~fk B " 


€iCO 



(8.104) 



and similar expressions for p > a. The fact that the fields arrange them- 
selves in two groups, (B p , Ej) depending on B z , and (B^, E p ) depending on 
E z , suggests that we attempt to obtain solutions of the TE or TM type, as 
for the metal wave guides. For the TE modes, the fields are explicitly 



B z = UYP) 

y 

cy 



p < a 



(8.105) 



262 

and 



Classical Electrodynamics 



B z = AK (P P ) 

p 



p > a 



(8.106) 



These fields must satisfy the standard boundary conditions at p = a. This 
leads to the two conditions, 

AK (fia) = J (ya) 



p y 



(8.107) 



Upon elimination of the constant A we obtain the determining equations 
for y, (3, and therefore k: 



Uya) | K^a) = Q 



(8.108) 



yJ (ya) ^K (^a) 
and, from (8.100) and (8.101), 

r 2 + ^ = ( ei -e )^ 
c l 

The general behavior of the two parts of the first equation in (8.108) is 
shown in Fig. 8.10a. Figure 8.106 shows the two curves superposed 




Pa 



(Ta)max = 




(a) 



(b) 



Fig. 8.10 Graphical determination of the axial propagation constant for a 
dielectric wave guide. 



[Sect. 8.8] Wave Guides and Resonant Cavities 263 

according to the second equation in (8.108). The frequency is assumed, to 
be high enough that two modes, marked by the circles at the intersections 
of the two curves, exist. The vertical asymptotes are given by the roots of 
J Q (x) = 0. If the maximum value of ya is smaller than the first root 
Ooi = 2.405), there can be no intersection of the two curves for real £. 
Hence the lowest "cutoff" frequency for TE n waves is given by 

2.405c 
w i = , (8.109) 

V e i _ e o a 

At this frequency £ 2 = 0, but the axial wave number k is still real and 
equal to its free-space value A/e w/c. Immediately below this "cutoff" 
frequency, the system no longer acts as a guide but as an antenna, with 
energy being radiated radially. For frequencies well above cutoff, ft and 
k are of the same order of magnitude and are large compared to y provided 
€ ± and e are not nearly equal. 
For TM modes, the first equation in (8.108) is replaced by 

yJ (ya) ^fiK^fia) V ' } 

It is evident that all the qualitative features shown in Fig. 8.10 are retained 
for the TM waves. The lowest "cutoff" frequency for TM „ waves is 
clearly the same as for TE >n waves. For e x > e , provided the maximum 
value of ya does not fall very near one of the roots of J (x) = 0, (8.110) 
shows that the propagation constants are determined by J x {ya) ~ 0. This 
is just the determining equation for TE waves in a metallic wave guide. 
The reason for the equivalence of the TM modes in a dielectric guide and 
the TE modes in a hollow metallic guide can be traced to the symmetry 
of Maxwell's equations under the interchange of E and B (with appro- 
priate sign changes and factors of Vjue), plus the correspondence between 
the vanishing of normal B at the metallic surface and the almost vanishing 
of normal E at the dielectric surface (due to continuity of normal D with 

If e x > e , then from (8.100) and (8.101) it is clear that the outside decay 
constant is much larger than y, except near cutoff. This means that the 
fields do not extend appreciably outside the dielectric cylinder. Figure 
8. 1 1 shows qualitatively the behavior of the fields for the TE 0jl mode. The 
other modes behave similarly. As mentioned earlier, modes with azimuthal 
dependence to the fields have longitudinal components of both E and B. 
Although the mathematics is somewhat more involved (see Problem 8.6), 
the qualitative features of propagation— short wavelength along the 
cylinder, rapid decrease of fields outside the cylinder, etc.— are the same 
as for the circularly symmetric modes. 



264 



Classical Electrodynamics 



a 






1 

1 

— ^ 1 






B, 
E 


B 


/b p 


X I 

\ 1 

\ 1 

\ 1 

\ 1 

\l 

E <t> \ 












1 
1 


p- 


— >- 



Fig. 8.11 Radial variation of 
fields of TE ,i mode in dielectric 
guide. For €i ^> e , the fields 
are confined mostly inside the 
dielectric. 



Dielectric wave guides have not been used for microwave propagation, 
except for special applications. One reason is that it is difficult to obtain 
suitable dielectrics with sufficiently low losses at microwave frequencies. 
In a recent application at optical frequencies very fine dielectric filaments, 
each coated with a thin layer of material of much lower index of refraction, 
are closely bundled together to form image-transfer devices.* The 
filaments are sufficiently small in diameter (~ 10 microns) that wave-guide 
concepts are useful, even though the propagation is usually a mixture of 
several modes. 



REFERENCES AND SUGGESTED READING 

Wave guides and resonant cavities are dealt with in numerous electrical and communi- 
cations engineering books. Among the physics textbooks which discuss guides, trans- 
mission lines, and cavities are 

Panofsky and Phillips, Chapter 12, 

Slater, 

Sommerfeld, Electrodynamics, Sections 22-25, 

Stratton, Sections 9.18-9.22. 
The mathematical tools for the discussion of these boundary-value problems are 
presented by 

Morse and Feshbach, especially Chapter 13. 
Information on special functions may be found in the ever-reliable 

Magnus and Oberhettinger. 
Numerical values of Bessel functions are given by 

Jahnke and Emde, 

Watson. 



PROBLEMS 

8.1 A transmission line consisting of two concentric circular cylinders of metal 
with conductivity a and skin depth <5, as shown on p. 265, is filled with a 

* B. O'Brien, Physics Today, 13, 52 (1960). 



[Probs. 8] 



Wave Guides and Resonant Cavities 



265 



uniform lossless dielectric (ji, e). A TEM mode is propagated along this line. 
(a) Show that the time-averaged power flow along the line is 



P = 



ira 2 \H n \ 2 In 



9 



where H is the peak value of the azimuthal magnetic field at the surface of 
the inner conductor. 




(6) Show that the transmitted power is attenuated along the line as 

P(z) =p e-2rz 



where 



2ct<5 



M , (b\ 
In I- I 



(c) The characteristic impedance Z of the line is defined as the ratio of 
the voltage between the cylinders to the axial current flowing in one of them 
at any position z. Show that for this line 



_ c _\2tt/\] e \aj 



(d) Show that the series resistance and inductance per unit length of the 
line are 



2tto6 \a bj 
2t7 \a/ 4tt \a b 



where n c is the permeability of the conductor. The correction to the 
inductance comes from the penetration of the flux into the conductors by a 
distance of order 8. 

8.2 A transmission line consists of two identical thin strips of metal, shown in 
cross section on p. 266. Assuming that b > a, discuss the propagation 



266 Classical Electrodynamics 

of a TEM mode on this line, repeating the derivations of Problem 8.1. 
Show that _ _ _ 

ab U. |2 









p 




c 

Att_ 

c 


v 




4tt 
4tt 


-o 




c 



aod/d ju 



Itia 
Ab 



R = 



2adb 



L = 



/ fig + /j, c d \ 
\ b J 



M J- 



where the symbols have the same meanings as in Problem 8.1. 

8.3 Transverse electric and magnetic waves are propagated along a hollow, 
right circular cylinder of brass with inner radius R. 

(a) Find the cutoff frequencies of the various TE and TM modes. Deter- 
mine numerically the lowest cutoff frequency (the dominant mode) in terms 
of the tube radius and the ratio of cutoff frequencies of the next four higher 
modes to that of the dominant mode. 

(b) Calculate the attenuation constant of the wave guide as a function of 
frequency for the lowest two modes and plot it as a function of frequency. 

8.4 A wave guide is constructed so that the cross_ section of the guide forms a 
right triangle with sides of length a, a, V2a, as shown on p. 267. The 
medium inside has /u = e = 1 . 



[Probs. 8] 



Wave Guides and Resonant Cavities 



267 



(a) Assuming infinite conductivity for the walls, determine the possible 
modes of propagation and their cutoff frequencies. 




(b) For the lowest modes of each type calculate the attenuation constant, 
assuming that the walls have large, but finite, conductivity. Compare the 
result with that for a square guide of side a made from the same material. 

8.5 A resonant cavity of copper consists of a hollow, right circular cylinder of 
inner radius R and length L, with flat end faces. 

(a) Determine the resonant frequencies of the cavity for all types of 
waves. With (c/V/x € R) as a unit of frequency, plot the lowest four resonant 
frequencies of each type as a function of RjL for < RjL < 2. Does the 
same mode have the lowest frequency for all RjLl 

(b) If R = 2 cm, L = 3 cm, and the cavity is made of pure copper, what 
is the numerical value of Q for the lowest resonant mode ? 

8.6 A right circular cylinder of nonpermeable dielectric with dielectric constant e 
and radius a serves as a dielectric wave guide in vacuum. 

(a) Discuss the propagation of waves along such a guide, assuming that 
the azimuthal variation of the fields is e im ^. 

(b) For m = ±1, determine the mode with the lowest cutoff frequency 
and discuss the properties of its fields (cutoff frequency, spatial variation, 
etc.), assuming that e > 1 . 



9 



Simple Radiating Systems 
and Diffraction 



In Chapters 7 and 8 we have discussed the properties of electro- 
magnetic waves and their propagation in both bounded and unbounded 
geometries. But nothing has been said about how to produce these waves. 
In the present chapter we remedy this omission to some extent by pre- 
senting a discussion of radiation by a localized oscillating system of 
charges and currents. The treatment is straightforward, with little in the 
way of elegant formalism. It is by its nature restricted to rather simple 
radiating systems. A more systematic approach to radiation by localized 
distributions of charge and current is left to Chapter 16, where multipole 
fields are discussed. 

The second half of the chapter is devoted to the subject of diffraction. 
Since the customary scalar Kirchhoff theory is discussed in many books, 
the emphasis has been placed on the vector properties of the electro- 
magnetic field in diffraction. 

9.1 Fields and Radiation of a Localized Oscillating Source 

For a system of charges and currents varying in time we can make a 
Fourier analysis of the time dependence and handle each Fourier com- 
ponent separately. We therefore lose no generality by considering the 
potentials, fields, and radiation from a localized system of charges and 
currents which vary sinusoidally in time : 

P (x, = P (x)e-^ (9 1} 

j(x, = J(x)e" J 
268 



[Sect. 9.1] Simple Radiating Systems and Diffraction 269 

As usual, the real part of such expressions is to be taken to obtain physical 
quantities. The electromagnetic potentials and fields are assumed to have 
the same time dependence. 

It was shown in Chapter 6 that the solution for the vector potential 
A(x, t) in the Lorentz gauge is 

A(x, = - fdV [df J(X '' ° d(t' + |x ~ x ' [ - t] (9.2) 

cJ J |x — x'| \ C I 

provided no boundary surfaces are present. The Dirac delta function 
assures the causal behavior of the fields. With the sinusoidal time 
dependence (9.1), the solution for A becomes 

I r e ik\x-x'\ 

A(x) = - J(x') <Pz' (9.3) 

cj |x — x I 

where k = ai/c is the wave number, and a sinusoidal time dependence is 
understood. The magnetic induction is given by 

B = V x A (9.4) 

while, outside the source, the electric field is 

E = -V x B (9.5) 

k 

Given a current distribution J(x'), the fields can, in principle at least, be 
determined by calculating the integral in (9.3). We will consider one or 
two examples of direct integration of the source integral in Section 9.4. 
But at present we wish to establish certain simple, but general, properties 
of the fields in the limit that the source of current is confined to a small 
region, very small in fact compared to a wavelength. If the source 
dimensions are of order d and the wavelength is X = Ittc/oj, and if d < X, 
then there are three spatial regions of interest: 

The near (static) zone : d < r < X 

The intermediate (induction) zone : d < r ^ X 
The far (radiation) zone : d < X < r 

We will see that the fields have very different properties in the different 
zones. In the near zone the fields have the character of static fields with 
radial components and variation with distance which depends in detail on 
the properties of the source. In the far zone, on the other hand, the fields 
are transverse to the radius vector and fall off as r _1 , typical of radiation 
fields. 



270 Classical Electrodynamics 

For the near zone where r « X (or kr « 1) the exponential in (9.3) can 
be replaced by unity. Then the vector potential is of the form already 
considered in Chapter 5. The inverse distance can be expanded using 
(3.70), with the result, 

lim A(x) = i 2 T^TZ ^T^ \mr' l Yti?, <f>') d 3 *' (9.6) 
fcr-o c t, to 2/ + 1 r + J 

This shows that the near fields are quasi-stationary, oscillating har- 
monically as e~ i(ot , but otherwise static in character. 

In the far zone {kr » 1) the exponential in (9.3) oscillates rapidly and 
determines the behavior of the vector potential. In this region it is 
sufficient to approximate 

|x — x'| ~ r — n • x' (9.7) 

where n is a unit vector in the direction of x. Furthermore, if only the 
leading term in kr is desired, the inverse distance in (9.3) can be replaced 
by r. Then the vector potential is 

lim A(x) = — J(x>~ ifcn - X d z x'. (9.8) 

at->oo cr J 

This demonstrates that in the far zone the vector potential behaves as an 

outgoing spherical wave. It is easy to show that the fields calculated 

from (9.4) and (9.5) are transverse to the radius vector and fall off as r~ x . 

They thus correspond to radiation fields. If the source dimensions are 

small compared to a wavelength it is appropriate to expand the integral in 

(9.8) in powers of A:: ikr / ;ky» f 

lim A(x) = — 2 — — J(x')(n • x') B d*x' (9.9) 

kr-+oo Cr n n\ J 

The magnitude of the nth term is given by 

J(x')(fcn • x') n d*x' (9.10) 



if. 



Since the order of magnitude of x' is d and kd is small compared to unity 
by assumption, the successive terms in the expansion of A evidently fall 
off rapidly with n. Consequently the radiation emitted from the source 
will come mainly from the first nonvanishing term in the expansion (9.9). 
We will examine the first few of these in the following sections. 

In the intermediate or induction zone the two alternative approxi- 
mations leading to (9.6) and (9.8) cannot be made; all powers of kr must 
be retained. Without marshalling the full apparatus of vector multipole 
fields, described in Chapter 16, we can abstract enough for our immediate 
purpose. The key result is the exact expansion (16.22) for the Green's 
function appearing in (9.3). For points outside the source (9.3) then 
becomes » 

A(*) = — 2 hf\kr)Y lm {B, +) J(x')j,(fcr')r,£(0', </>') d 3 x' (9.11) 

C 7. TO J 



Simple Radiating Systems and Diffraction 271 

If the source dimensions are small compared to a wavelength, j t {kr') can 
be approximated by (16.12). Then the result for the vector potential is of 
the form of (9.6), but with the replacement, 

^1 "* ^1 0- + a & kr> > + a ^ ikr ? +■'• + ^{ikrf ) (9.12) 

The numerical coefficients a t come from the explicit expressions for the 
spherical Hankel functions. The right hand side of (9.12) shows the 
transition from the static-zone result (9.6) for kr « 1 to the radiation-zone 
form (9.9) for kr » 1. 

9.2 Electric Dipole Fields and Radiation 

If only the first term in (9.9) is kept, the vector potential is 

ikr f 

A(x) = 6 — J J(x') c/V (9.13) 

Examination of (9.11) and (9.12) shows that (9.13) is the / = part of 
the series and that it is valid everywhere outside the source, not just in the 
far zone. The integral can be put in more familiar terms by an integration 
by parts : 

J J d z x' = - J x'(V . J) d z x' = -ico\x'p(x') d 3 x' (9.14) 

from the continuity equation, 

iojp = V.j 
Thus the vector potential is ., v ' } 

r p tkr 

where A(x)=-f/cp- (9 16) 

P=jx' P (x')d 3 x' (9.17) 

is the electric dipole moment, as defined in electrostatics by (4.8). 
The electric dipole fields from (9.4) and (9.5) are 



B = k 2 (n x p)— (l - — 
r \ ikr 



E = k\n x p) x n e ~ + [3n(n • p) - p](i - ^), 



(9.18) 



We note that the magnetic induction is transverse to the radius vector at 
all distances, but that the electric field has components parallel and perpen- 
dicular to n. 

In the radiation zone the fields take on the limiting forms, 

a ikr\ 

B = fc 2 (n x p) — 

E = B x n J 

showing the typical behavior of radiation fields. 



272 Classical Electrodynamics 

In the near zone, on the other hand, the fields approach 



B = ik(n x p) - 
r 1 

E=[3n(n.p)-p]i 



(9.20) 



The electric field, apart from its oscillations in time, is just the static 

electric dipole field (4.13). The magnetic induction is a factor (kr) smaller 

than the electric field in the region where kr < 1 . Thus the fields in the 

near zone are dominantly electric in nature. The magnetic induction 

vanishes, of course, in the static limit k -> 0. Then the near zone extends 

to infinity. 

The time-averaged power radiated per unit solid angle by the oscillating 

dipole moment p is 

dP c 

^- = ±- Re [r 2 n . E x B*] (9.21) 

dQ. 8-7T 

where E and B are given by (9.19). Thus we find 

dP c 

^ = - fc*|n x (n x p)| 2 (9.22) 

dil 67T 

The state of polarization of the radiation is given by the vector inside the 
absolute value signs. If the components of p all have the same phase, the 
angular distribution is a typical dipole pattern, 

^ = f fc 4 |p| 2 sin 2 (9.23) 

dil 57T 

where the angle 6 is measured from the direction of p. The total power 
radiated is 

P = C -f IPI 2 (9.24) 

A simple example of an electric dipole radiator is a centerfed, linear 
antenna whose length d is small compared to a wavelength. The antenna 
is assumed to be oriented along the z axis, extending from z = (d/2) to 
z — —(d/2) with a narrow gap at the center for purposes of excitation, as 
shown in Fig. 9.1. The current is in the same direction in each half of the 
antenna, having a value I at the gap and falling approximately linearly to 
zero at the ends : 

I(z)e- i0)t = lj\ - 'M\e- i0>t (9.25) 



[Sect. 9.3] 



Simple Radiating Systems and Diffraction 



273 




Fig. 9.1 Short, center-fed, linear antenna. 

From the continuity equation (9.15) the linear-charge density p (charge 
per unit length) is constant along each arm of the antenna, with the value, 



p'(*) = ± 



2i7 fl 



(9.26) 



the upper (lower) sign being appropriate for positive (negative) values of z. 
The dipole moment (9.17) is parallel to the z axis and has the magnitude 



-r 



zp'(z) dz= 1 -^ 

(d/2) 2a> 



The angular distribution of radiated power is 
dP V 



dQ. 32-7TC 
while the total power radiated is 

P 



{kdf sin 2 



I \kdf 



12c 



(9.27) 



(9.28) 



(9.29) 



We see that for a fixed input current the power radiated increases as the 
square of the frequency, at least in the long-wavelength domain where 
kd <1. 

9.3 Magnetic Dipole and Electric Quadrupole Fields 

The next term in expansion (9.9) leads to a vector potential, 

A(x) = — (- - ikj j* J(x')(n • x') dV (9.30) 

where we have included the correct terms from (9.12) in order 
to make (9.30) valid everywhere outside the source. This vector 
potential can be written as the sum of two terms, one of which 
gives a transverse magnetic induction and the other of which gives a 



274 Classical Electrodynamics 

transverse electric field. These physically distinct contributions can be 
separated by writing the integrand in (9.30) as the sum of a part symmetric 
in J and x' and a part that is antisymmetric. Thus 

- (n • x')J = — [(n • x')J + (n • J)x'] + — (x' x J) x n (9.31) 

c 2c 2c 

The second, antisymmetric part is recognizable as the magnetization due 
to the current J: 

M = — (x x J) (9.32) 

2c 

The first, symmetric term will be shown to be related to the electric 
quadrupole moment density. 

Considering only the magnetization term, we have the vector potential, 

A(x) = ik(n x m) — ( 1 - — ) (9.33) 

r \ ikrf 

where m is the magnetic dipole moment, 

m = \jl d z x = — (x x J) d 3 x (9.34) 

The fields can be determined by noting that the vector potential (9.33) is 
proportional to the magnetic induction (9.18) for an electric dipole. This 
means that the magnetic induction for the present magnetic dipole source 
will be equal to the electric field for the electric dipole, with the substitution 
p —>■ m. Thus we find 

B = k 2 (n x m) x n — + [3n(n • m) - m](- - - )e ikr (9.35) 
r \r 3 r 2 / 

Similarly, the electric field for a magnetic dipole source is the negative of 
the magnetic field for an electric dipole : 

E = -k\n x m)— (l - — ) (9.36) 

r \ ikrf 

All the arguments concerning the behavior of the fields in the near and 
far zones are the same as for the electric dipole source, with the inter- 
changes E->B, B— >- — E, p^-m. Similarly the radiation pattern and 
total power radiated are the same for the two kinds of dipole. The only 
difference in the radiation fields is in the polarization. For an electric 
dipole the electric vector lies in the plane defined by n and p, while for a 
magnetic dipole it is perpendicular to the plane defined by n and m. 



[Sect. 9.3] Simple Radiating Systems and Diffraction 275 

The integral of the symmetric term in (9.31) can be transformed by an 
integration by parts and some rearrangement : 

- f [(n • x')J + (n • J)x'] d*x' = - |fx'(n • x>(x') d 3 x' (9.37) 

The continuity equation (9.15) has been used to replace V • J by imp. 
Since the integral involves second moments of the charge density, this 
symmetric part corresponds to an electric quadrupole source. The vector 
potential is 

A(x) = - - — (l - — ) f x'(n • x>(x') d 3 x' (9.38) 

2 r \ ikrf J 

The complete fields are somewhat complicated to write down. We will 
content ourselves with the fields in the radiation zone. Then it is easy to 
see that 

B = ikn x A 1 

(9.39) 
E = ik(n x A) x n J 

Consequently the magnetic induction is 

* i 3 Hct /* 
B = - — — (n x x')(n • x>(x') d 3 x' (9.40) 

2 r J 

With definition (4.9) for the quadrupole moment tensor, 

Q* P = J(3*.*, - ^) P (x) d*x (9.41) 

the integral in (9.40) can be written 

n x J x'(n • x')p(x') d 3 x' = ^n x Q(n) (9.42) 

The vector Q(n) is defined as having components, 

<2a = I<Vv (9-43) 

p 

We note that it depends in magnitude and direction on the direction of 
observation as well as on the properties of the source. With these defi- 
nitions we have the magnetic induction, 

■r.3 Jkr 

B = - — — n x Q(n) (9.44) 

6 r 

and the time-averaged power radiated per unit solid angle, 

— = — /c 6 [n x Q(n)| 2 (9.45) 

dQ. 288tt 



276 



Classical Electrodynamics 



The general angular distribution is complicated. But the total power 
radiated can be calculated in a straightforward way. With the definition of 
Q(n) we can write the angular dependence as 



|n x Q(n)| 2 = Q* • Q - |n • Q| 2 

= J, Q*pQ« Y n p n y - 2 Q*pQydn a n p n y n d 



<*>/?, y,<5 



(9.46) 



The necessary angular integrals over products of the rectangular com- 
ponents of n are readily found to be 



l 



n R n y dQ 



I 



4-7T 



477 



J fiv 



n a n p n y n d dQ. = — (d af} d yd + d ay d p5 + d ad d Py ) 



(9.47) 



Then we find 



/' 



n x Q(n)| 2 dQ = AnU 2 \Q a(S \ 2 - A 2 Ql lQyy + ^2 \<L, 



*,p 



a,P 



(9.48) 

Since Q afi is a tensor whose main diagonal sum is zero, the first term in the 
square brackets vanishes identically. Thus we obtain the final result for 
the total power radiated by a quadrupole source: 



'-£2iw 



(9.49) 



The radiated power varies as the sixth power of the frequency for fixed 
quadrupole moments, compared to the fourth power for dipole radiation. 
A simple example of a radiating quadrupole source is an oscillating 
spheroidal distribution of charge. The off-diagonal elements of Q aj3 vanish. 
The diagonal elements may be written 



(?33 — Qo-> Qn — Q22 — — \Qo 
Then the angular distribution of radiated power is 
dP c/c 6 



dQ. 128tt 



Q ' sin 2 6 cos 2 



(9.50) 



(9.51) 



This is a four-lobed pattern, as shown in Fig. 9.2, with maxima at 6 = tt/4 
and 377-/4. The total power radiated by this quadrupole is 



P = 



240 



(9.52) 



[Sect. 9.4] 



Simple Radiating Systems and Diffraction 



111 



Fig. 9.2 Quadrupole radiation pattern. 




The labor involved in manipulating higher terms in expansion (9.9) of 
the vector potential (9.8) becomes increasingly prohibitive as the expansion 
is extended beyond the electric quadrupole terms. Another disadvantage 
of the present approach is that physically distinct fields such as those of the 
magnetic dipole and the electric quadrupole must be disentangled from 
the separate terms in (9.9). Finally, the present technique is useful only in 
the long- wavelength limit. A systematic development of multipole radia- 
tion is given in Chapter 16. It involves a fairly elaborate mathematical 
apparatus, but the price paid is worth while. The treatment allows all 
multipole orders to be handled in the same way; the results are valid for 
all wavelengths ; the physically different electric and magnetic multipoles 
are clearly separated from the beginning. 



9.4 Center-fed Linear Antenna 



For certain radiating systems the geometry of current flow is sufficiently 
simple that integral (9.3) for the vector potential can be found in relatively 
simple, closed form. As an example of such a system we consider a thin, 
linear antenna of length d which is excited across a small gap at its mid- 
point. The antenna is assumed to be oriented along the z axis with its gap 
at the origin, as indicated in Fig. 9.3. If damping due to the emission of 
radiation is neglected, the current along the antenna can be taken as 
sinusoidal in time and space with wave number k = co/c, and is symmetric 
on the two arms of the antenna. The current vanishes at the ends of the 



278 Classical Electrodynamics 

antenna. Hence the current density can be written 

J(x) = /sin(^-/c| Z |)^)%)€ 3 



(9.53) 



for \z\ < (d/2). The delta functions assure that the current flows only 
along the z axis. / is the peak value of the current if kd > tt. The current 
at the gap is I = I sin (kd/2). 

With the current density (9.53) the vector potential is in the z direction 
and in the radiation zone has the form [from (9.7)] : 



Ie 1 ' 



— ikz cos 6 j 

e dz 



kt , „ r f W2) . (kd .. A 
A(x) = e 3 sin — - k \z\ I 

cr J-(,d/2) \ 2 1 

The result of straightforward integration is 

(| cose) -cos (I) 



(9.54) 



A(x) = €, 



2Ie l 



ckr - 



cos 



sin 2 



(9.55) 



Since the magnetic induction in the radiation zone is given by B = 
ikn x A, its magnitude is |B| = k sin 6 \A 3 \. Thus the time-averaged power 
radiated per unit solid angle is 

(kd A (kd\ 2 

cos 



dp 
da 



JL 

2ttc 



(| cose) -cos (|) 



sin 6 



(9.56) 



The electric vector is in the direction of the component of A perpendicular 
to n. Consequently the polarization of the radiation lies in the plane 
containing the antenna and the radius vector to the observation point. 



Coaxial 
feed 




Fig. 9.3 Center-fed, linear antenna. 



[Sect. 9.4] 



Simple Radiating Systems and Diffraction 



279 



The angular distribution (9.56) depends on the value of kd. In the 
long- wavelength limit {kd < 1) it is easy to show that it reduces to the 
dipole result (9.28). For the special values kd = tt (2tt), corresponding to 
a half (two halves) of a wavelength of current oscillation along the antenna, 
the angular distributions are 



dP 
dQ. 



JL 

lire 



cos 



(fees,) 



sin 



4 cos 4 I - cos 
\2 

sin 2 



kd = tt 



kd = 2TT 



(9.57) 



These angular distributions are shown in Chapter 16 in Fig. 16.4, where 
they are compared to multipole expansions. The half-wave antenna 
distribution is seen to be quite similar to a simple dipole pattern, but the 
full-wave antenna has a considerably sharper distribution. 

The full-wave antenna distribution can be thought of as due to the 
coherent superposition of the fields of two half-wave antennas, one above 
the other, excited in phase. The intensity at 6 = -njl, where the waves add 
algebraically, is 4 times that of a half- wave antenna. At angles away from 
6 = tt/2 the amplitudes tend to interfere, giving the narrower pattern. By 
suitable arrangement of a set of basic antennas, such as the half-wave 
antenna, with the phasing of the currents appropriately chosen, arbitrary 
radiation patterns can be formed by coherent superposition. The interested 
reader should refer to the electrical engineering literature for detailed 
treatments of antenna arrays. 

For the half-wave and full-wave antennas the angular distributions can 
be integrated over angles to give 



P = '- 

c 



kd = tt 



(9.58) 



1 P'/l - cos A . 

2 Jo \-^) dt ' 

The integrals in (9.58) can be expressed in terms of the cosine integral, 

Ci(x)= - f°°2°il^ (9.59) 

J X t 

f x / l - cos t \ df = ln ^ x) _ a ^ (9 6Q) 



as follows : 



280 Classical Electrodynamics 

where y = 1.781 . . , is Euler's constant. Tables of the cosine integral 

are given by Jahnke and Emde, pp. 6-9. The numerical results for the 

power radiated are 

72 (2.44, kd = tt 
P = - (9.61) 

2c (6.70, kd = 2tt 

For a given peak current / the full-wave, center-fed antenna radiates 
nearly 3 times as much power as the half-wave antenna. The coefficient 
of P/2 has the dimensions of a resistance and is called the radiation 
resistance R Tad of the antenna. The value in ohms is obtained from (9.61) 
by multiplying the numbers by 30 (actually the multiplier is the numerical 
value of the velocity of light divided by appropriate powers of 10). Thus 
the half- and full-wave center-fed antennas have radiation resistances of 
73.2 ohms and 201 ohms, respectively. 

The reader should be warned that the idealized problem of an infinitely 
thin, linear antenna with a sinusoidal current distribution is a somewhat 
simplified version of what occurs in practice. Finite lateral dimensions, 
ohmic and radiative losses, nonsinusoidal current distributions, finite gaps 
for excitation, etc., all introduce complications. These problems are 
important in practical applications and are treated in detail in an extensive 
literature on antenna design, to which the interested reader may refer. 



9.5 Kirchhoff's Integral for Diffraction 

The general problem of diffraction involves a wave incident on one or 
more obstacles or apertures in absorbing or conducting surfaces. The 
wave is scattered and perhaps absorbed, leading to radiation propagating 
in directions other than the incident direction. The calculation of the 
radiation emerging from a diffracting system is the aim of all diffraction 
theories. The earliest systematic attempt was that of G. Kirchhoff (1882), 
based on the ideas of superposition of elemental wavelets due to Huygens. 
In this section we will discuss Kirchhoff's method and point out some of 
its deficiencies, and in the next section derive vector theorems which 
correspond to the basic scalar theorem of Kirchhoff. 

The customary geometrical situation in diffraction is two spatial regions 
I and II separated by a boundary surface S, as shown in Fig. 9.4. For 
example, S may be an infinite metallic sheet with certain apertures in it. 
The incident wave, generated by sources in region I, approaches S from 
one side and is diffracted at the boundary surface, giving rise to scattered 
waves, one transmitted and one reflected. It is usual to consider only the 
transmitted wave and call its distribution in angle the diffraction pattern 



[Sect. 9.5] 



Simple Radiating Systems and Diffraction 



281 



E r , B r 



E t , B( 



Eo,Bo 



■*■ E 0) Bo 



Fig. 9.4 Diffracting system. The surface 
S, with certain apertures in it, gives rise to 
reflected and transmitted fields in regions I 
and II in addition to the fields which would 
be present in the absence of the surface. 

of the system. If the incident wave is described by the fields E , B , the 
reflected wave by the fields E r , B r , and the transmitted wave by E t , B t , 
then the total fields in regions I and II are E = E + E s , B = B + B s , 
where s stands for r or /. The basic problem is to determine (E t , B t ) and 
(E r , B r ) from the incident fields (E , B ) and the properties of the boundary 
surface S. To connect the fields in region I with those in region II 
boundary conditions for E and B must be satisfied on S, the form of these 
boundary conditions depending on the properties of S. 

The method of attack used in solving such problems is the Green's 
theorem technique, as applied to the wave equation in Chapter 6. Con- 
sider a scalar field ip(x, t) defined on and inside a closed surface S and 
satisfying the source-free wave equation in that region. The field ^(x, t) 
can be thought of as a rectangular component of E or B. We proved in 
Chapter 6 that the value of ip inside S could be written in terms of the value 
of ip and its normal derivative on the surface as 



4tt JsR 



R 



V>(x', t') - £ <Kx', O - — 



R ay>(x', Q" 



n da' 



R 2 cR dt' -U 

(9.62) 

where R = x — x', n is the outwardly directed normal to the surface, and 
ret means evaluated at a time t' = t — (R/c). If a harmonic time depen- 
dence e~ ia * is assumed, this integral form for ip(x, i) can be written: 

ikR 



w(x) = — <P n 

4ttJs R 



<t — n- V>+ifc(l + — |-v da' (9.63) 

Is R I \ kR/R J 



To adapt (9.63) to diffraction problems we consider the closed surface 
S to be made up of two surfaces S 1 and S 2 . Surface 5 X will be chosen as a 
convenient one for the particular problem to be solved (e.g., the con- 
ducting screen with apertures in it), while surface S 2 will be taken as a 
sphere or hemisphere of very large radius (tending to infinity) in region II, 
as shown in Fig. 9.5. Since the fields in region 11 are the transmitted fields 



282 



Classical Electrodynamics 



which originate from the diffracting region, they will be outgoing waves in 
the neighborhood of S 2 . This means that the fields, and therefore ip(x), 
will satisfy the radiation condition, 



V-/(M) 



1 dtp 
tp dr 



'*-; 



(9.64) 



With this condition on tp it can readily be seen that the integral in (9.63) 
over the hemisphere S 2 vanishes inversely as the hemisphere radius as that 
radius goes to infinity. Then we obtain the Kirchhoff integral for tp(x) in 
region II : 



y<x) 



= _J_f £!!_„. v>+ ikll + — )-tp da' (9.65) 
4ttJ Si R i \ kR/R J 



where n is now a unit vector normal to S x and pointing into region II. 

In order to apply the Kirchhoff formula (9.65) to a diffraction problem 
it is necessary to know the values of tp and dtp/dn on the surface S v Unless 
we have already solved the problem exactly, these values are not known. 
If, for example, S ± is a plane, perfectly conducting screen with an opening 
in it and tp represents the component of electric field parallel to S v then we 
know that tp vanishes everywhere on S x , except in the opening. But the 
value of tp in the opening is undetermined. Without additional knowledge, 
only approximate solutions can be found by making some assumption 
about tp and dtp/dn on S v The Kirchhoff approximation consists of the 
assumptions : 

1 . tp and dtp/dn vanish everywhere on S t except in the openings. 

2. The values of tp and dtp/dn in the openings are equal to the values of 
the incident wave in the absence of any screens or obstacles. 

The standard diffraction calculations of classical optics are all based on the 
Kirchhoff approximation. It should be obvious that the recipe can have 
only very approximate validity. There is a basic mathematical incon- 
sistency in the assumptions. It was shown for Laplace's equation (and 
equally well for the Helmholtz wave equation) in Section 1.9 that the 



Sources 





Fig. 9.5 Possible diffraction 
geometries. Region I contains 
the sources of radiation. Region 
II is the diffraction region, where 
the fields satisfy the radiation 
condition. 



[Sect. 9.6] Simple Radiating Systems and Diffraction 283 

solution inside a closed volume is determined uniquely by specifying ip 
(Dirichlet boundary condition) or dip/dn (Neumann boundary condition) 
on the surface. Both ip and dtp/dn cannot be given on the surface. The 
Kirchhoff approximation works best in the short-wavelength limit in 
which the diffracting openings have dimensions large compared to a wave- 
length. Being a scalar theory, even there it cannot account for details of 
the polarization of the diffracted radiation. In the intermediate- and 
long-wavelength limit, the scalar approximation fails badly, aside from the 
drastic approximations inherent in the basic assumptions listed above. 

Since the diffraction of electromagnetic radiation is a boundary-value 
problem in vector fields, we expect that a considerable improvement can 
be made by developing vector equivalents to the Kirchhoff integral (9.65). 

9.6 Vector Equivalents of Kirchhoflf Integral 

To obtain vector equivalents to the Kirchhoff integral (9.63) we first 
note that with the definition, 

G(x,x') = - — (9.66) 

477 R 

the scalar form (9.63) can be written 

w (x) = i> [Gn • V> - yn • V'G] da' (9.67) 

By writing down the result (9.67) for each rectangular component of the 
electric or magnetic field and combining them vectorially, we can obtain 
the vector theorem, 

E(x) = <f> [G(n • V')E - E(n • V'G)] da' (9.68) 

with a corresponding relation for B. This result is not a particularly 
convenient one for calculations. It can be transformed into a more useful 
form by a succession of vector manipulations. First the integrand in 
(9.68) can be written 

[ ] = (n • V')(GE) - 2E(n • V'G) (9.69) 

Then the vector identities, 

n x (E x V'G) = E(n • V'G) - (n • E)V'G 1 

V'G x (n x E) = n(E • V'G) - E(n • V'G) J 
can be combined to eliminate the last term in (9.69) : 

[ ] = (n . V')(GE) - n x (E x V'G) - n(E • V'G) 

-(n • E)V'G - (n x E) x V'G (9.71) 



284 Classical Electrodynamics 

Now the curl of the product of a vector and a scalar is used to transform 
the second term in (9.71), while the fact that V • E = is used to re- 
express the third term. The result is 

[ ] = ( n • V')(GE) + n x V x (GE) - nV • (GE) 

-(n • E)V'G - (n x E) x V'G - Gn x (V x E) 

(9.72) 

While it may not appear very fruitful to transform the two terms in (9.68) 
into six terms, we will now show that the surface integral of the first three 
terms in (9.72), involving the product (GE), vanishes identically. To do 
this we make use of the following easily proved identities connecting 
surface integrals over a closed surface S to volume integrals over the interior 
of S: 



i> A • n da = V • A d 3 x 

Js Jv 

(p(nxA)Ja= V x A d 3 
Js Jv 

\ <f>n da = V<f> cPx 

Js Jv 



(9.73) 



where A and <f> are any well-behaved vector and scalar functions. With 
these identities the surface integral of the first three terms in (9.72) can be 
written 

<j> [(n • V')(GE) + n x V x (GE) - nV • (GE)] da' 

= [V' 2 (GE) + V x V x (GE) - V'(V • (GE))] dV (9.74) 

Jv 

From the expansion, V x V x A = V(V • A) — V 2 A, it is evident that 
the volume integral vanishes identically. * 

With the surface integral of the first three terms in (9.72) identically 
zero, the remaining three terms give an alternative form for the vector 
Kirchhoff relation (9.68). From Maxwell's equations we have V x E = 
ikB, so that the final result for the electric field anywhere inside the volume 

* The reader may well be concerned that theorems (9.73) do not apply, since the 
vector function (GE) is singular at the point x' = x. But if the singularity is excluded by 
taking the surface S as an outer surface 5" and a small sphere S" around x' = x, the con- 
tribution of the integral over S" can be shown to vanish in the limit that the radius of 
S" goes to zero. Hence result (9.74) is valid, even though G is singular inside the volume 
of interest. 



[Sect. 9.6] Simple Radiating Systems and Diffraction 285 

bounded by the surface S is 

E(x) = - <j> [i/c(n x B)G + (n x E) x V'G + (n • E)V'G] da' (9.75) 

The analogous expression for the magnetic induction is 

B(x) = - &> l-ik(n x E)G + (n x B) x V'G + (n • B)V'G] da' (9.76) 

In (9.75) and (9.76) the unit vector n is the usual outwardly directed normal. 
These integrals have an obvious interpretation in terms of equivalent 
sources of charge and current. The normal component of E in (9.75) is 
evidently an effective surface-charge density. Similarly, according to 
(8.14), the tangential component of magnetic induction (n x B) acts as an 
effective surface current. The other terms (n • B) and (n x E) are effective 
magnetic surface charge and current densities, respectively. 

Vector formulas (9.75) and (9.76) serve as vector equivalents to the 
Huygens-Kirchhoff scalar integral (9.63). If the fields E and B are assumed 
to obey the radiation condition (9.64) with the added vectorial relationship, 
E = B x (r/r), it is easy to show that the surface integral at infinity 
vanishes. Then, in the notation of Fig. 9.5, the electric field (9.75) is 

E(x) = [(n x E) x V'G + (n • E)V'G + ik(n x B)G] da' (9.77) 

where S x is the surface appropriate to the diffracting system, and n is now 
directed into the region of interest. 

The vector theorem (9.77) is a considerable improvement over the 
scalar expression (9.65) in that the vector nature of the electromagnetic 
fields is fully included. But to calculate the diffracted fields it is still 
necessary to know the values of E and B on the surface S v The Kirchhoff 
approximations of the previous section can be applied in the short-wave- 
length limit. But the sudden discontinuity of E and B from the unperturbed 
values in the "illuminated" region to zero in the "shadow" region on the 
back side of the diffracting system must be compensated for mathemati- 
cally by line currents around the boundaries of the openings. * 

A very convenient formula can be obtained from (9.77) for the special 
case of plane boundary surface S v We imagine that the surface S ± 
containing the sources in the right-hand side of Fig. 9.5 is changed in 
shape into a large, flat pancake, as shown in Fig. 9.6. The region II of 
"transmitted" fields now becomes two regions, II and II', connected 
together only by an annular opening at infinity. We denote the two sides 

* For a discussion of these line currents, see Stratton, pp. 468-470, and Silver, 
Chapter 5. 



286 



Classical Electrodynamics 




Fig. 9.6 

of the disc by S x and S±. The unit vectors n and n' = — n are directed into 
regions II and II', respectively. Our aim is to obtain an integral form for 
the fields in region II in terms of the fields specified on the right-hand 
surface S x . This is analogous to the geometrical situation shown in the 
left side of Fig. 9.5. We do not care about the values of the fields in region 
II'. In fact, the hypothetical sources inside the disc will be imagined to be 
such that the fields in region II' give a contribution to the surface integral 
(9.77) which makes the final expression for the diffracted fields in region II 
especially useful. Once we have obtained the desired result [equation 
(9.82) below] for the fields in region II as an integral over the surface S x , 
we will forget about the manner of derivation and ignore the whole left- 
hand side of Fig. 9.6. Our interest is in the diffracted fields in region II 
caused by apertures or obstacles located on the plane surface S v 

If the fields in regions II and II' are E, B and E', B', respectively, then 
from the figure it is evident that when the thickness of the disc becomes 
vanishingly small, integral (9.77) may be written 

E(x) = J [(n x (E - E')) x V'G + n • (E - E')V'G 

+ ikn x (B - B')G] da' (9.78) 
The field E(x) on the left side is either E or E', depending on where the 
point x lies. But the integral is over the right-hand surface S x only. 
One of the most common applications is to conducting surfaces with 
apertures in them. The boundary conditions at a perfectly conducting 
surface are n x E = 0, n • B = 0, but n • E =£ 0, n x B ^ 0. In cal- 
culating the surface integral in (9.78) it would be desirable to integrate 
only over the apertures in the surface rather than over all of it. The first 



[Sect. 9.6] Simple Radiating Systems and Diffraction 287 

term in (9.78) exists only in the apertures if the screen is perfectly con- 
ducting. Consequently we try to choose the fields in region II' so that the 
other terms vanish everywhere on S v Evidently we must choose 

(n • E%' = (n • E) Sl } 

(9.79) 
(n x B%; = (n x B) Sl J 

Of course, the fields E', B' must satisfy Maxwell's equations and the 
radiation condition in region II' if E, B satisfy them in region II. It is easy 
to show that the required relationship, giving (9.79) on the surfaces, is 

n x E'(x') = -n x E(x) 

n • E'(x') = n • E(x) 
n x B'(x') = n x B(x) (9 * 80) 

n • B'(x') = -n • B(x) 

where the point x' is the mirror image of x in the plane S v The fields at 
mirror-image points have the opposite (same) values of tangential and 
outwardly directed normal components of electric field (magnetic 
induction). 

With conditions (9.80) in (9.78) we obtain the simple result for the field 
E(x) in terms of an integral over the plane surface S x bounding region II,* 

E(x) = 2 (n x E) x V'G da' (9.81) 

where (n x E) is the tangential electric field on S lt n is a unit normal 
directed into region II, and G is the Green's function (9.66). Since 
V = —V when operating on G, (9.81) can be put in the alternate form, 

E(x) = 2V x | n x E(x')G(x, x') da' (9.82) 

For a diffraction system consisting of apertures in a perfectly conducting 
plane screen the integral over S t may be confined to the apertures only. 
Result (9.81) or (9.82) is exact if the correct tangential component of E 
over the apertures is inserted. In practice, we must make some approxi- 
mation as to the form of the aperture field. But, for plane conducting 
screens at least, only the tangential electric field need be approximated 
and the boundary conditions on the screen are correctly satisfied [as can 
be verified explicitly from (9.82)]. 

* This form for plane screens was first obtained by W. R. Smythe, Phys. Rev., 72, 
1066 (1947), using an argument based on the fields due to a double current sheet filling 
the apertures, rather than the present Green's-theorem technique. 



288 



Classical Electrodynamics 



9.7 Babinet's Principle of Complementary Screens 

Before discussing examples of diffraction we wish to establish a useful 
relation called Babinet's principle. Babinet's principle relates the dif- 
fraction fields of one diffracting screen to those of the complementary 
screen. We first discuss the principle in the scalar Kirchhoff approxi- 
mation. The diffracting screen is assumed to lie in some surface S which 
divides space into regions I and II in the sense of Section 9.5. The screen 
occupies all of the surface S except for certain apertures. The comple- 
mentary screen is that diffracting screen which is obtained by replacing 
the apertures by screen and the screen by apertures. If the surface of the 
original screen is S a and that of the complementary screen is S b , then 
S a + $b = S, as shown schematically in Fig. 9.7. 

If there are sources inside S (in region I) which give rise to a field y(x), 
then in the absence of either screen the field ip(x) in region II is given by 
the Kirchhoff integral (9.65) where the surface integral is over the entire 
surface S. With the screen S a in position, the field y> a (x) in region II is 
given in the Kirchhoff approximation by (9.65) with the source field y) in 
the integrand and the surface integral only over S b (the apertures). 
Similarly, for the complementary screen S b , the field ip b (x) is given in the 
same approximation by a surface integral over S a . Evidently, then, we 
have the following relation between the diffraction fields tp a and \p b : 



Wa + Wb = W 



(9.83) 



This is Babinet's principle as usually formulated in optics. If ip represents 
an incident plane wave, for example, Babinet's principle says that the 





S a 



Fig. 9.7 A diffraction screen S a and its 
complementary diffraction screen S b - 



[Sect. 9.7] Simple Radiating Systems and Diffraction 289 

diffraction pattern away from the incident direction is the same for the 
original screen and its complement. 

The above formulation of Babinet's principle is unsatisfactory in two 
aspects : it is a statement about scalar fields, and it is based on the Kirchhoff 
approximation. The second deficiency can be remedied by defining the 
complementary problem as not only involving complementary screens 
but also involving complementary boundary conditions (Dirichlet versus 
Neumann) for the scalar fields. But since we are interested in the electro- 
magnetic field, we will not pursue the scalar problem further. 

A rigorous statement of Babinet's principle for electromagnetic fields 
can be made for a thin, plane, perfectly conducting screen and its comple- 
ment. We start by considering certain fields E , B incident on the screen 
with metallic surface S a (see Fig. 9.7) in otherwise empty space. The 
presence of the screen gives rise to transmitted and reflected fields, as 
shown in Fig. 9.4. These transmitted and reflected fields will be denoted 
collectively as scattered fields, E s , B s , unless we need to be more specific. For 
a perfectly conducting screen, the surface current K induced by the incident 
fields must be such that at all points on the screen's surface S a , n x E g = 
— n x E . For a thin, plane surface, the symmetry of the problem implies 
that the tangential components of scattered magnetic field at the surface 
must be equal and opposite, being given from (5.90) by 

n x H t = — K = -n x H r (9.84) 

c 

where n points into the transmitted region II. As a matter of fact, by the 
same arguments that led from (9.79) to (9.80), it can be established that 
at any point x in region II and its mirror-image point x' in region I, the 
scattered fields satisfy the symmetry conditions, 



n x E r (x') = n x E ( (x) 
n • E r (x') = -n • E t (x) 

n x B r (x') = — n x B 4 (x) 
nB r (x') = nB t (x) 



(9.85) 



It will be noted that these relations differ from those in (9.80) by having 
the signs of E r (x') and B r (x') reversed. As we see from the work of Smythe 
(pp. cit., Section 9.6), the fields of (9.80) correspond to a double layer of 
current. The present fields have the symmetries (9.85) appropriate to a 
single, plane, current sheet radiating in both directions. 

An integral expression for the scattered magnetic induction can now be 
written down in terms of the surface current K. Since B is the curl of the 



290 Classical Electrodynamics 

vector potential, we have 

B s = V x — I KG da' (9.86) 

c Js a 

where G is the Green's function (9.66), and the integration goes over the 
metallic surface S a of the screen. If we substitute for K from (9.84), we 
can write the magnetic induction in region II as 

B t (x) = 2V x n x B 4 (x')G(x, x') da (9.87) 

Js a 

This result is identical with (9.82) except that 

(1) the roles of E and B have been interchanged, 

(2) the present integration is only over the body of the screen, whereas that 
in (9.82) is only over the apertures, 

(3) the total electric field appears in (9.82), whereas only the scattered 
fields occur in (9.87). 

The comparison of (9.87) with (9.82) forms the basis of Babinet's 
principle. If we write down the result (9.82) for the complement of the 
screen with metallic surface S a , we have 

E'(x) = 2V x n x E'(x')G(x, x') da' (9.88) 

'So 

The integration is only over S a , since that is the aperture in the comple- 
mentary screen. The field E' in region II is the sum, 

E' = E ' + E/ (9.89) 

where E ' is the incident electric field of the complementary diffraction 

problem, and E/ the corresponding transmitted or diffracted field. 

Evidently the two expressions (9.87) and (9.88) turn into one another under 

the transformation, 

B,->±(E ' + E/) (9.90) 

It is easy to show that the other fields transform at the same time according 

E t ^T(B ' + B/) (9.91) 

the sign difference arising from the fact that the fields must represent 
outgoing radiation in both cases. Since we could have started with the 
complementary screen initially, it is clear that (9.90) and (9.91) must hold 
equally with the primed and unprimed quantities interchanged. Com- 
parison of the two sets of expressions shows that the incident fields of the 
original and complementary diffraction problems must be related accord- 

mgt ° E '=-B , B ' = E (9.92) 

The complementary problem involves not only the complementary screen, 



[Sect. 9.7] Simple Radiating Systems and Diffraction 291 



Fig. 9.8 Equivalent radiators according to Babinet's principle. 

but also a complementary set of incident fields with the roles of E and B 
interchanged. 

The statement of Babinet's principle is therefore as follows: a dif- 
fracting system consists of a source producing fields E , B incident on a 
thin, plane, perfectly conducting screen with certain apertures in it. The 
complementary diffracting system consists of a source producing fields 
E ' = — B , B ' = E incident on the complementary screen. If the 
transmitted (diffraction) fields on the opposite side of the screens from the 
source are E ( , B t and E/, B/ for the diffracting system and its complement, 
respectively, then they are related by 

E< + B/ = — E = — B ' 



(9.93) 
B« — E/ = — B = +E ' 

These are the vector analogs of the scalar relation (9.83). 

If a plane wave is incident on the diffracting screen, Babinet's principle 
states that, in directions other than the incident direction, the intensity of 
the diffraction pattern of the screen and its complement will be the same, 
the fields being related by 

E* = — B/ 



B, = E/ 



(9.94) 



The polarization of the wave incident on the complementary screen must, 
of course, be rotated according to (9.92). 

The rigorous vector formulation of Babinet's principle is very useful in 
microwave problems. For example, consider a narrow slot cut in an 
infinite, plane, conducting sheet and illuminated with fields that have the 
magnetic induction along the slot and the electric field perpendicular to 
it, as shown in Fig. 9.8. The radiation pattern from the slot will be the 
same as that of a thin linear antenna with its driving electric field along the 
antenna, as considered in Sections 9.2 and 9.4. The polarization of the 
radiation will be opposite for the two systems. Elaboration of these ideas 
makes it possible to design antenna arrays by cutting suitable slots in the 
sides of wave guides.* 

* See, for example, Silver, Chapter 9. 



292 Classical Electrodynamics 



9.8 Diffraction by a Circular Aperture 

The subject of diffraction has been extensively studied since Kirchhoff 's 
original work, both in optics, where the scalar theory based on (9.65) 
generally suffices, and in microwave generation and transmission, where 
more accurate solutions are needed. There exist specialized treatises 
devoted entirely to the subject of diffraction and scattering. We will 
content ourselves with a few examples to illustrate the use of the scalar 
and vector theorems (9.65) and (9.82) and to compare the accuracy of the 
approximation schemes. 

Historically, diffraction patterns were classed as Fresnel diffraction and 
Fraunhofer diffraction, depending on the distance of the observation point 
from the diffracting system. Generally the diffracting system (e.g., an 
aperture in an opaque screen) has dimensions comparable to, or large 
compared to, a wavelength. Then the observation point may be in the 
near zone, less than a wavelength away from the diffracting system. The 
near-zone fields are complicated in structure and of little interest. Points 
many wavelengths away from the diffracting system, but still near the 
system in terms of its own dimensions, are said to lie in the Fresnel zone. 
Further away, at distances large compared to both the dimensions of 
the diffracting system and the wavelength, lies the Fraunhofer zone. The 
Fraunhofer zone corresponds to the radiation zone of Section 9. 1 . The 
diffraction patterns in the Fresnel and Fraunhofer zones show character- 
istic differences which come from the fact that for Fresnel diffraction the 
region of the diffracting system nearest the observation point is of greatest 
importance, whereas for Fraunhofer diffraction the whole diffracting 
system contributes. We will consider only Fraunhofer diffraction, leaving 
examples of Fresnel diffraction to the problems at the end of the chapter. 

If the observation point is far from the diffracting system, expansion 
(9.7) can be used for R = |x — x'|. Keeping only lowest-order terms in 
(1/Ar), the scalar Kirchhoff expression (9.65) becomes 



= - e —[ 
4irr Js 



y(x) = - — \ e 

477T J Si 



-tk-x' 



n • V'y(x') + ik • n^(x') 



da' (9.95) 



where x' is the coordinate of the element of surface area da', r is the length 
of the vector x from the origin O to the observation point P, and k = 
k(x/r) is the wave vector in the direction of observation, as indicated in 
Fig. 9.9. For a plane surface the vector expression (9.82) reduces in this 
limit to 

ie ikr f t < 

E(x) = — k x n x E(x>~ Jk ' x da' (9.96) 

2irr JSi 



[Sect. 9.8] 



Simple Radiating Systems and Diffraction 



293 




Fig. 9.9 

As an example of diffraction we consider a plane wave incident at an 
angle a on a thin, perfectly conducting screen with a circular hole of radius 
a in it. The polarization vector of the incident wave lies in the plane of 
incidence. Figure 9.10 shows an appropriate system of coordinates. The 
screen lies in the x-y plane with the opening centered at the origin. The 
wave is incident from below, so that the domain z > is the region of 
diffraction fields. The plane of incidence is taken to be the x-z plane. The 
incident wave's electric field, written out explicitly in rectangular com- 
ponents, is , x ifc(cosaz + sin as) (Q Qn\ 
E; = E (e x cos a — € 3 sin a)e (9.97) 

In calculating the diffraction field with (9.95) or (9.96) we will make the 
customary approximation that the exact field in the surface integral may 
be replaced by the incident field. For the vector relation (9.96) we need 

(n x E^ =0 = £ e 2 cos xe ik sina *' (9.98) 

Then, introducing plane polar coordinates for the integration over the 
opening, we have 



E(x) = 



ie ikr E cos a 



(k x € 2 ) 



pdp\ c 
o Jo 



dpe 



t'fcpfsinacos/3— sin0cos(<£— /?)] 



2t7T * ~J ' 'Jo ' (9.99) 

where 6, <f> are the spherical angles of k. If we define the angular function, 
I = (sin 2 + sin 2 a - 2 sin 6 sin a cos <f>)* (9.100) 



Fig. 9.10 Diffraction by a circu- 
lar hole of radius a. 




294 Classical Electrodynamics 

the angular integral can be transformed into 

— \ dp = —\ d^e~ ap6coaP ' = J (k P £) (9.101) 

Itt Jo 2tt Jo 

Then the radial integral in (9.99) can be done directly. The resulting 
electric field in the vector Kirchhoff approximation is 



E(x) = a 2 E cos a(k x e 2 ) 



Ji(ka£) 



r ka£ 

The time-averaged diffracted power per unit solid angle is 



— = p i cos a — — (cos 2 <f> + cos 2 6 sin 2 <f>) 
dLl 4tt 



2/ 1 (fca£) 



where 



/ca| 



P, 



Tra cos a 



(9.102) 



(9.103) 



(9.104) 



is the total power normally incident on the aperture. If the opening is large 
compared to a wavelength {ka > 1), the factor [2J x {kak)lkai\ 2 peaks 
sharply to a value of unity at £ = and falls rapidly to zero (with small 
secondary maxima) within a region A£ ~ (l/ka) on either side of £ = 0. 
This means that the main part of the wave passes through the opening in 
the manner of geometrical optics; only slight diffraction effects occur. 
For ka ~ 1 the Bessel-function varies comparatively slowly in angle; the 
transmitted wave is distributed in directions very different from the 
incident direction. For ka < 1, the angular distribution is entirely deter- 
mined by the factor (k x e 2 ) in (9.102). But in this limit the assumption of 
an unperturbed field in the aperture breaks down badly. 

The total transmitted power can be obtained by integrating (9.103) over 
all angles in the forward hemisphere. The ratio of transmitted power to 
incident power is called the transmission coefficient T: 



cos a C 2 * C" /2 
T= cosa ^ (cos 2 <^ + cos 2 sin 2 0) 

TV Jo Jo 



J^kaZ) 



sin Odd (9.105) 



In the two extreme limits ka > 1 and ka < 1, the transmission coefficient 

approaches the values, 

f cos a, ka > 1 

T^ I (9.106) 

[ %{kdf cos a, ka < 1 

The long- wavelength limit {ka < 1) is suspect because of our approxi- 
mations, but it shows that the transmission is small for very small holes. 



[Sect. 9.8] Simple Radiating Systems and Diffraction 295 

For normal incidence (a = 0) the transmission coefficient (9.105) can be 
written 

'2 



T=\ J t \ka sin 0)(- 

Jo \sm 

With the help of the integral relations, 



- sin dd 



(9.107) 



pr/2 

Jo ' 



J n %z sin 0) 



dd 



sin 



f«r/2 

Jo ' 



J n \z sin 0)sin dd 



Jo * 



df 



(9.108) 



and the recurrence formulas (3.87) and (3.88), the transmission coefficient 
can be put in the alternative forms, 



T = 



1 °° 
1 - — ^Jzm+iilka) 

TO = 

-i /*2fco 

1-rf- Ut)dt 

2ka Jo 



(9.109) 



The transmission coefficient increases more or less monotonically as ka 
increases, with small oscillations superposed. For ka > 1, the second form 
in (9.109) can be used to obtain an asymptotic expression, 



T~ 1 - 



1 



1 



2ka 2jir{kay 



sin 



( 2k "-l) 



+ 



(9.110) 



which exhibits the small oscillations explicitly. These approximate expres- 
sions (9.109) and (9.110) for Tgive the general behavior as a function of 
ka, but are not very accurate. Exact calculations, as well as more accurate 
approximate ones, have been made for the circular opening. These are 
compared with each other in the book by King and Wu (Fig. 41, p. 126). 
The correct asymptotic expression does not contain the l/2ka term in 
(9.110), and the coefficient of the term in (ka)- 3A is twice as large. 

We now wish to compare our results of the vector Kirchhoff approxi- 
mation with the usual scalar theory based on (9.95). For a wave not 
normally incident the question immediately arises as to what to choose for 
the scalar function y(x). Perhaps the most consistent assumption is to 
take the magnitude of the electric or magnetic field. Then the diffracted 
intensity is treated consistently as proportional to the absolute square of 
(9.95). If a component of E or B is chosen for ip, we must then decide 
whether to keep or throw away radial components of the diffracted field in 



296 



Classical Electrodynamics 



calculating the diffracted power. Choosing the magnitude of E for tp, 
we have, by straightforward calculation with (9.95), 



ip(x) = —ik — a 2 E \ 



2l7 /cos a + cos 6 \ J x (kaEi) 
2 / kaS 



(9.111) 



as the scalar equivalent of (9. 102). The power radiated per unit solid angle 
in the scalar Kirchhoff approximation is 



dP _ (kaf 
— ~ P, v — - cos 
dQ. 4tt 



/cos a + cos 6\ 2 
\ 2 cos a / 



2Ji(fca^) 



ka£ 



(9.112) 



where P t is given by (9.104). 

If we compare the vector Kirchhoff result (9.103) with (9.112), we see 
similarities and differences. Both formulas contain the same "diffraction" 
distribution factor [J 1 (kag)/kag] 2 and the same dependence on wave 
number. But the scalar result has no azimuthal dependence (apart from 
that contained in |), whereas the vector expression does. The azimuthal 
variation comes from the polarization properties of the field, and must be 
absent in a scalar approximation. For normal incidence (a = 0) and 
ka > 1 the polarization dependence is unimportant. The diffraction is 




(a) 





/ 


\ 


/ 


\ 


1 


\ 


1 


\ 


I 


' 


)/' 


\ , 


1/ 


\\ 


// 


\\ 


// 


\ 


/ 


R * 



(b) 



Fig. 9.11 Fraunhofer diffraction pattern for a circular opening one wavelength in 
diameter in a thin, plane, conducting sheet. The plane wave is incident on the screen 
at 45°. The solid curves are the vector Kirchhoff approximation, while the dotted curves 
are the scalar approximation, (a) The intensity distribution in the plane of incidence 
(E plane), (b) The intensity distribution (enlarged 2.5 times) perpendicular to the plane 

of incidence (H plane). 



[Sect. 9.9] Simple Radiating Systems and Diffraction 297 

confined to very small angles in the forward direction. Then both scalar 
and vector approximations reduce to the common expression, 



dP ^ p (kaf 
dQ. l IT 



J x (ka sin 6) 



ka sin 6 



(9.113) 



The vector and scalar Kirchhoff approximations are compared in Fig. 
9.11 for the angle of incidence equal to 45° and for an aperture one wave- 
length in diameter (ka — n). The angular distribution is shown in the plane 
of incidence (containing the electric field vector of the incident wave) and a 
plane perpendicular to it. The solid (dotted) curve gives the vector (scalar) 
approximation in each case. We see that for ka = tt there is a considerable 
disagreement between the two approximations. There is reason to believe 
that the vector Kirchhoff result is close to the correct one, even though the 
approximation breaks down seriously for ka < 1. The vector approxi- 
mation and exact calculations for a rectangular opening yield results in 
surprisingly good agreement, even down to ka ~ \* 



9.9 Diffraction by Small Apertures 

In the large-aperture or short-wavelength limit we have seen that a 
reasonably good description of the diffracted fields is obtained by approxi- 
mating the tangential electric field in the aperture by its unperturbed 
incident value. For longer wavelengths this approximation begins to fail. 
When the apertures have dimensions small compared to a wavelength, an 
entirely different approach is necessary. We will consider a thin, fiat, 
perfectly conducting sheet with a small hole in it. The dimensions of the 
hole are assumed to be very small compared to a wavelength of the electro- 
magnetic fields which are assumed to exist on one side of the sheet. The 
problem is to calculate the diffracted fields on the other side of the sheet. 
Since the sheet is assumed flat, the simple vector theorem (9.82) is appro- 
priate. Evidently the problem is solved if we can determine the electric 
field in the plane of the hole. 

As pointed out by Bethe (1942), the fields in the neighborhood of the 
aperture can be treated by static or quasi-static methods. In the absence 
of the aperture the electromagnetic fields near the conducting plane 
consist of a normal electric field E and a tangential magnetic induction 
B on one side, and no fields on the other. By "near the conducting plane," 
we mean at distances small compared to a wavelength. If a small hole is 

* See J. A. Stratton and L. J. Chu, Phys. Rev., 56, 99 (1939), for a series of figures 
comparing the vector Kirchhoff approximation with exact calculations by P. M. Morse 
and P. J. Rubenstein, Phys. Rev., 54, 895 (1938). 



298 



Classical Electrodynamics 

Eq 





Fig. 9.12 

now cut in the plane, the fields will be altered and will penetrate through 
the hole to the other side. But far away from the hole (in terms of its 
dimensions), although still "near the conducting plane," the fields will be 
the same as if the hole were not there, namely, normal E and tangential 
B . The electric field lines might appear as shown in Fig. 9.12. Since the 
departures of the fields E and B from their unperturbed values E and B 
occur only in a region with dimensions small compared to a wavelength, 
the task of determining E or B near the aperture becomes a problem in 
electrostatics or magnetostatics, apart from the overall sinusoidal time 
dependence e~ imt . For the electric field, it is a standard potential problem 
of knowing the "asymptotic" values of E on either side of the perfectly 
conducting sheet which is an ftquipotential surface. Similarly for the 
magnetic induction, B must be found to yield B and zero "asymptotically" 
on either side of the sheet, with no normal component on the surface. Then 
the electric field due to the time variation of B can be calculated and 
combined with the "electrostatic" electric field to give the total electric 
field near the opening. 

For a circular opening of radius a small compared to a wavelength, for 
example, the tangential electric field in the plane of the opening can be 

shown to be . 

E tan = E J. + — (n x B )Va 2 - P 2 (9.114) 

W« 2 — p* "" 
where E = E • n is the magnitude of the normal electric field in the 
absence of the hole, B is the tangential magnetic induction in the absence 
of the hole, n is the unit vector normal to the surface and directed into the 
diffraction region [as in (9.82)], and p is the radius vector in the plane 
measured from the center of the opening. With this tangential field deter- 
mined in the static limit it is a straightforward matter to determine the 



[Sect. 9.10] Simple Radiating Systems and Diffraction 299 

diffracted fields and power from (9.82). The calculations for the circular 
opening will be left to the problems at the end of the chapter (Problems 
9.10 and 9.11). 



9.10 Scattering by a Conducting Sphere in the 
Short- Wavelength Limit 

Another type of problem which is essentially diffraction is the scattering 
of waves by an obstacle. We will consider the scattering of a plane 
electromagnetic wave by a perfectly conducting obstacle whose dimensions 
are large compared to a wavelength. For a thin, flat obstacle, the tech- 
niques of Section 9.8, perhaps with Babinet's principle, can be used. But 
for other obstacles we base the calculation on vector theorem (9.77) for 
the scattered fields. If we consider only the fields in the radiation zone 
(kr > 1), the integral (9.77) for the scattered field E s becomes 

E s — — f [(n x E s ) x k + (n • E s )k - fc(n x B s )>- ik * x ' da' 
4-rrir Jst 

(9.115) 

where k is the wave vector of the scattered wave, and S x is the surface of 
the obstacle. It will be somewhat easier to calculate with the magnetic 
induction B s = (k x E s )/fc: 



B s -^ — k xj f(n x E s ) x 7 -n x B, 



— ik-x' 



v „ .. „ e~ l *-*da' (9.116) 

4mr Jsi L k 

In the absence of knowledge about the correct fields E s and B s on the 
surface of the obstacle, we must make some approximations. If the wave- 
length is short compared to the dimensions of the obstacle, the surface 
can be divided approximately into an illuminated region and a shadow 
region. * The boundary between these regions is sharp only in the limit of 
geometrical optics. The transition region can be shown to have a width of 
the order of (2lkR) 1A R, where R is a typical radius of curvature of the 
surface. Since R is of the order of magnitude of the dimensions of the 
obstacle, the short-wavelength limit will approximately satisfy the geo- 
metrical condition. In the shadow region the scattered fields on the surface 
must be very nearly equal and opposite to the incident fields. In the 
illuminated region, the scattered tangential electric field and normal 
magnetic induction must be equal and opposite to the corresponding 
incident fields in order to satisfy the boundary conditions on the surface 

* For a very similar treatment of the scattering of a scalar wave by a sphere, see Morse 
and Feshbach, pp. 1551-1555. 



300 



Classical Electrodynamics 



of the perfectly conducting obstacle. On the other hand, the tangential B s 
and normal E s in the illuminated region will be approximately equal to 
the incident values, just as for an infinite, flat, conducting sheet, to the 
extent that the wavelength is small compared to the radius of curvature. 
Thus we obtain the following approximate values for the scattered fields 
on the surface of the obstacle : 



Shadow Region 

E s ~-E; 

B. ~ -B, 



Illuminated Region 
n x E s = — n x E, 

n • B s = -n • B, 
n x B s ~ n x B^ 

n • E ~ n • E, 



where E i5 B^ are the fields of the incident wave. With these boundary 
values the scattered magnetic induction (9.116) can be written as 

ATcr 



B a 



Airir 



k x (F sh + Fin) 



where 



'sh 



JshL/c 



x E,) + n x B, 



— ik-x' 



da' 



is the integral over the shadow region, and 



Fin = I 

Jil 



-x(nxEi)-nxBi 
k 



-ik-x' 



da 



(9.117) 
(9.118) 

(9.119) 



is the integral over the illuminated region. 
If the incident wave is a plane wave with wave vector k^ 

E/x) = E e ik °-* \ 

B,(x) = ^° x E,(x) j 



(9.120) 



the integrals over the shadow and illuminated regions of the obstacle's 
surface are 



F sh = J- f [(k + ko) x (n x E ) + (n . E )k o y' (k °- k) - x 'rfa' 

Fill = r I [(k - ko) x (n x E ) - (n . E )k ]e , " (k °- k) - 1 ' da' 
k Jill 



(9.121) 



These integrals behave very differently as functions of the scattering angle. 
In the short-wavelength limit the magnitudes of k • x' and k • x' are large 
compared to unity. Thus the exponential factors in (9.121) will oscillate 



[Sect. 9.10] Simple Radiating Systems and Diffraction 301 

rapidly and cause the integrands to have very small average values except 
in the forward direction where k ^ 1^. In that direction the second term 
in both F sh and F m is unimportant, since the scattered field (9.117) is 
proportional to k x F. The behavior of the two contributions is thus 
governed by the first terms in (9.121), at least in the forward direction. We 
see that F sh and F m are proportional to (k ± 1%), respectively; the 
shadow integral will be large and the integral from the illuminated region 
will go to zero. As the scattering angle departs from the forward direction 
the shadow integral will vanish rapidly, both the exponential and the 
vector factor in the integrand having the same tendency. On the other 
hand, the integral from the illuminated region will be small in the forward 
direction and can be expected to be small at all angles, the exponential and 
the vector factor in the integrand having opposite tendencies. The shadow 
integral is evidently the diffraction contribution, while the integral from 
the illuminated region is the reflected wave. 

To proceed much further we must specify the shape of the obstacle. We 
will assume that it is a perfectly conducting sphere of radius a. Since the 
shadow integral is large only in the forward direction, we will evaluate it 
approximately by placing k = k^, everywhere except in the exponential. 
Then, omitting the second term in (9.121) and using spherical coordinates 
on the surface of the sphere, we obtain 



J'ff/2 f2ir 

„•„ a „~„ iA"a(l— cos 0)cosa I in - 
sin a d<x cos cue dp e 

o Jo 



, . r^. _ ,„ „- . c,„ „ „~ ™c ~ „ • . -z'fca sine sin a cos (/?-(£) 



(9.122) 

The angles 6, </> and a, /3 are those of k and n relative to 1^. The exponential 
factor involving (1 — cos 6) can be set equal to unity, since at small angles 
its exponent is a factor 0/2 smaller than the other exponent. The integral 
over /? is 2rrJ (ka sin 6 sin a). Hence 



"sh ~ — 



r*/2 

47ra 2 E I J (kad sin a)cos a sin a dot. (9.123) 

Jo 

where we have approximated sin 6 ~ 6. The integral over a is pro- 

fkaO 

portional to the integral xJ Q {x) dx = kad J^kad). Therefore the shadow 

cr'Cttte^rincr intorrrol if •'0 



scattering integral is 



F sh --47r 2 aE ^^ (9.124) 

kad 



We see that this is essentially the diffraction field of a circular aperture 
(9.102). 

The integral over the illuminated region, giving the reflected or back- 
scattered wave, is somewhat harder to evaluate. We must consider 



302 Classical Electrodynamics 

arbitrary scattering angles, since there is no enhancement in the forward 
direction. Then the integral consists of a relatively slowly varying vector 
function of angles times a rapidly varying exponential. As is well known, 
the dominant contribution to such an integral comes from the region of 
integration where the phase of the exponential is stationary. The phase 
factor is 

/(a, /?) = (k„ — k) • x' = ka[(l — cos 0) cos a — sin 6 sin a cos (/? — </>)] 

(9.125) 
The stationary point is easily shown to be at angles Oq, /? , where 

77 0) 

2 2 (9.126) 

These angles are evidently just those appropriate for reflection from the 
sphere according to geometrical optics. At this point the unit vector n 
points in the direction of (k — Ilq). If we expand the phase factor around 
a = a , /? = ($ , we obtain 

1 - - (x 2 + cos 2 - y 2 ) + • • 



/(a, {$) = —2ka sin - 



2 \ 2 



(9.127) 



where x = a — olq, y = /S — /9 . Then integral (9.121) can be approxi- 
mated by evaluating the square bracket at a = a , /? = /? : 

F„i ^ a 2 sin 0[2(n o • E ) n - E >- 2itosin W2) 

x \dx e i[ka sin m)]x2 \dy e i[ka sin (e/2) cos2 (e/2)]2/2 (9 128) 

where n,, is a unit vector in the direction (k — k ). Provided 6 is not too 
small, the phase factors oscillate rapidly for large x or y. Hence the 
integration can be extended to ± oo in each integral without error. Using 
the result, 



we obtain 



J e Ux * dx= (-) e iw/ * (9.129) 



Fiu ^ i ^ e~ 2ikasin (fl/2) [2(n • EoK - E ] (9.130) 

k 



After some vector algebra the contribution to the scattered field from the 
illuminated part of the sphere can be written 



,ifcr 



E ( s iU) ^ - - £ — e~ 2ika sin W 2 > e m (9.131) 

2 r 



[Sect. 9.10] 



Simple Radiating Systems and Diffraction 



303 




Fig. 9.13 Polarization of reflected 
wave relative to the incident polari- 
zation. 

where the polarization vector e m has a direction denned in Fig. 9.13. If 
the polarization vector of the incident wave E makes an angle d with the 
normal to the plane containing the wave vectors k and k„, the azimuthal 
angle y of e m , measured from the plane containing k and ko, is given by 
y _ (y/2) _ <5. We note that the reflected field (9.131) is constant in 
magnitude as a function of angle, although it has a rapidly varying 
phase. 
The scattered electric field due to the shadow region is, from (9.124) and 

(9.117), 

E (sh) ^ ika 2 JJM1 g!!( k x e ) x k (9 132) 

s had r k 2 

Comparison of the two contributions to the scattered wave shows that in 
the forward direction the shadow field is larger by a factor ka > 1. But 
for angles much larger than ~ (l/ka) the shadow field becomes very 
small and the isotropic reflected field dominates. The power scattered per 
unit solid angle can be expressed in the form: 



dP* 
dQ 



~ PA 



(kaf 
Att 

_1_ 

47T ' 



U^kad) 



kaO 



< 



ka 



(9.133) 



ka 



where P t = (c£ 2 a 2 /8) is the incident power per unit area times the pro- 
jected area (-rra 2 ) of the sphere. At small angles the scattering is a typical 
diffraction pattern [see (9.1 13)]. At large angles the scattering is isotropic. 
At intermediate angles the two amplitudes interfere, causing the scattered 
power to have sharp minimum values considerably smaller than the 



304 



Classical Electrodynamics 




Fig. 9.14 Diffraction pattern for 
a conducting sphere, snowing the 
forward peak due to shadow 
scattering, the isotropic reflected 
contribution, and the interference 
maxima and minima. 



isotropic value at certain angles, as shown in Fig. 9.14. The amount of 
interference depends on the orientation of the incident polarization vector 
relative to the plane of observation containing k and k . For E in this 
plane the interference is much greater than for E perpendicular to it.* 

The total power scattered is obtained by integrating over all angles. 
Neglecting the interference terms, the total scattered power is the sum of 
the integrals of the diffraction peak and the isotropic reflected part. The 
integrals are easily shown to be equal in magnitude. Hence 



P g = P, + i>< = IP, 



(9.134) 



We sometimes rephase this result by saying that the effective area of the 
sphere for scattering (its scattering cross section) is IttcP. One factor of 
77-a 2 comes from the direct reflection; the other comes from the diffraction 
scattering which must accompany the formation of a shadow behind the 
obstacle. 

Scattering of electromagnetic waves by a conducting sphere is treated 
by another method, especially in the long-wavelength limit, in Section 16.9. 



REFERENCES AND SUGGESTED READING 



The simple theory of radiation from a localized source distribution is discussed in all 
modern textbooks. Treatments analogous to that given here may be found in 
Panofsky and Phillips, Chapter 13, 
Stratton, Chapter VIII. 

* See King and Wu, Appendix, for numerous graphs of scattering by spheres as a 
function of ka. 



[Probs. 9] Simple Radiating Systems and Diffraction 305 

More complete discussions of antennas and antenna arrays are given in engineering 
works, such as 

Jordan, 

Kraus, 

Schelkunoff, 

Silver. 
The subject of diffraction has a very extensive literature. A comprehensive treatment 
of both the scalar Kirchhoff and the vector theory, with many examples and excellent 
figures, is given by 

Born and Wolf, Chapters VIII, IX, and XI. 
A more elementary discussion of the scalar theory is found in 

Slater and Frank, Chapters XIII and XIV. 
Mathematical techniques for diffraction problems are discussed by 

Baker and Copson, 

Morse and Feshbach, Chapter 11. 
The vector theorems used in Sections 9.6-9.10 are presented by 

Morse and Feshbach, Chapter 13, 

Silver, Chapter 5, 

Stratton, Sections 8.14 and 8.15. 
A specialized monograph on the scattering of electromagnetic waves by obstacles, with 
an emphasis on useful numerical results, is the book by 

King and Wu. 



PROBLEMS 

9.1 Discuss the power flow and energy content of the complete electric dipole 
fields (9.18) in terms of the complex Poynting's vector S = (c/8tt)(E x B*) 
and the time-averaged energy density u = (1/16tt)(E • E* + B • B*). The 
real part of S gives the true, resistive power flow, while the imaginary part 
represents circulating, reactive power. 

(a) Show that the real part of S is in the radial direction and is given by 
r~~ 2 times equation (9.23). 

(b) Show that the imaginary part of S has components in the r and 
directions given by 

ImS r = ^|p| 2 sin 2 

lmS = - ^- 2 (1 + A: 2 /- 2 ) sin B cos 6 

Make a sketch to show the direction of circulating power flow by suitably 
oriented arrows, the length of each arrow being proportional to the 
magnitude of Im S at that point. 

(c) Calculate the time-averaged energy density: 

1 |3n(np) -p| 2 /c 2 |n • p| a A: 4 [n x p] 2 
" ~ 16tt r 6 + 4nr* + Sirr* 

(d) Derive Poynting's theorem for the complex Poynting's vector. To 
what is Im (V • S) equal? Verify that this holds true for the results of (b) 
and (c). 



306 Classical Electrodynamics 

9.2 A radiating quadrupole consists of a square of side a with charges ±q at 
alternate corners. The square rotates with angular velocity co about an 
axis normal to the plane of the square and through its center. Calculate 
the quadrupole moments, the radiation fields, the angular distribution of 
radiation, and the total radiated power in the long-wavelength approxi- 
mation. 

9.3 Two halves of a spherical metallic shell of radius R and infinite conductivity 
are separated by a very small insulating gap. An alternating potential is 
applied between the two halves of the sphere so that the potentials are 
±V cos at. In the long-wavelength limit, find the radiation fields, the 
angular distribution of radiated power, and the total radiated power from 
the sphere. 

9.4 A thin linear antenna of length d is excited in such a way that the sinusoidal 
current makes a full wavelength of oscillation as shown in the figure. 



(a) Calculate exactly the power radiated per unit solid angle and plot 
the angular distribution of radiation. 

(b) Determine the total power radiated and find a numerical value for 
the radiation resistance. 

9.5 Treat the linear antenna of Problem 9.4 by the long-wavelength multipole 
expansion method. 

(a) Calculate the multipole moments (electric dipole, magnetic dipole, 
and electric quadrupole). 

(b) Compare the angular distribution for the lowest nonvanishing 
multipole with the exact distribution of Problem 9.4. 

(c) Determine the total power radiated for the lowest multipole and the 
corresponding radiation resistance. 

9.6 A perfectly conducting flat screen occupies one-half of the x-y plane 
(i.e., x < 0). A plane wave of intensity I and wave number k is incident 
along the z axis from the region z < 0. Discuss the values of the diffracted 
fields in the plane parallel to the x-y plane defined by z — Z > 0. Let the 
coordinates of the observation point be (X, 0, Z). 

(a) Show that, for the usual scalar Kirchhoff approximation and in the 
limit Z > X, the diffracted field is 

Y(X, 0, Z) =* i^gikz-iwtl 1 + A jl ("%«"«/, 

where £ = (k/2Z)^X. 

(b) Show that the intensity can be written 



/ = 



&(c(4) + i) 2 + m) + m 



where C(f) and S(€) are the so-called Fresnel integrals. Determine the 
asymptotic behavior of I for £ large and positive (illuminated region) and I 



[Probs. 9] Simple Radiating Systems and Diffraction 307 

large and negative (shadow region). What is the value of / at X = ? 
Make a sketch of / as a function of X for fixed Z. 

(c) Use the vector formula (9.82) to obtain a result equivalent to that of 
part (a). Compare the two expressions. 

9.7 A linearly polarized plane wave of amplitude E and wave number k is 
incident on a circular opening of radius a in an otherwise perfectly con- 
ducting flat screen. The incident wave vector makes an angle a with the 
normal to the screen. The polarization vector is perpendicular to the plane 
of incidence. 

{a) Calculate the diffracted fields and the power per unit solid angle 
transmitted through the opening, using the vector Kirchhoff formula 
(9.82) with the assumption that the tangential electric field in the opening 
is the unperturbed incident field. 

(b) Compare your result in part (a) with the standard scalar Kirchhoff 
approximation and with the result in Section 9.8 for the polarization 
vector in the plane of incidence. 

9.8 A rectangular opening with sides of length a and b > a defined by 
x = ±(a/2), y = ±(6/2) exists in a flat, perfectly conducting plane sheet 
filling the x-y plane. A plane wave is normally incident with its polarization 
vector, making an angle yS with the long edges of the opening. 

{a) Calculate the diffracted fields and power per unit solid angle with the 
vector Kirchhoff relation (9.82), assuming that the tangential electric field 
in the opening is the incident unperturbed field. 

(b) Calculate the corresponding result of the scalar Kirchhoff approxi- 
mation. 

(c) For b = a, P = 45°, ka = 4n, compute the vector and scalar approxi- 
mations to the diffracted power per unit solid angle as a function of the 
angle 6 for <f> = 0. Plot a graph showing a comparison between the two 
results. 

9.9 A cylindrical coaxial transmission line of inner radius a and outer radius b 
has its axis along the negative z axis. Both inner and outer conductors end 
at z = 0, and the outer one is connected to an infinite plane flange occupy- 
ing the whole x-y plane (except for the annulus of radius b around the 
origin). The transmission line is excited at frequency o> in its dominant 
TEM mode, with the peak voltage between the cylinders being V. Use the 
vector Kirchhoff approximation to discuss the radiated fields, the angular 
distribution of radiation, and the total power radiated. 

9.10 Discuss the diffraction due to a small, circular hole of radius a in a flat, 
perfectly conducting sheet, assuming that ka < 1. 

(a) If the fields near the screen on the incident side are normal E e -ltof 
and tangential B e- iait , show that the diffracted electric field in the 
Fraunhofer zone is 



a ikr —ieot 

E = k 2 a 3 

lirr 



2*x 

k 



b ° + H e ° x 9" 



where k is the wave vector in the direction of observation. 

(b) Determine the angular distribution of the diffracted radiation and 
show that the total power transmitted through the hole is 



308 Classical Electrodynamics 

9.11 Specialize the discussion of Problem 9.10 to the diffraction of a plane wave 
by the small, circular hole. Treat the general case of oblique incidence at an 
angle a to the normal, with polarization in and perpendicular to the plane 
of incidence. 

(a) Calculate the angular distributions of the diffracted radiation and 
compare them to the vector Kirchhoff approximation results of Section 9.8 
and Problem 9.7 in the limit ka < 1 . 

(b) Show that the transmission coefficients [defined above (9.105)] for 
the two states of polarization are 



T„ = 



64 

27tt 2 



(W /4+sin*a\ 
\ 4 COS a / 



64 

Th 2 



T L = ^-2 (kaf cos a 



Note that these transmission coefficients are a factor (ka) 2 smaller than 
those given by the vector Kirchhoff approximation in the same limit. 



10 



Magnetohydrodynamics 
and Plasma Physics 

10.1 Introduction and Definitions 

Magnetohydrodynamics and plasma physics both deal with the 
behavior of the combined system of electromagnetic fields and a con- 
ducting liquid or gas. Conduction occurs when there are free or quasi-free 
electrons which can move under the action of applied fields. In a solid 
conductor, the electrons are actually bound, but can move considerable 
distances on the atomic scale within the crystal lattice before making 
collisions. Dynamical effects such as conduction and Hall effect are 
observed when fields are applied to the solid conductor, but mass motion 
does not in general occur. The effects of the applied fields on the atoms 
themselves are taken up as stresses in the lattice structure. For a fluid, on 
the other hand, the fields act on both electrons and ionized atoms to 
produce dynamical effects, including bulk motion of the medium itself. 
This mass motion in turn produces modifications in the electromagnetic 
fields. Consequently we must deal with a complicated coupled system of 
matter and fields. 

The distinction between magnetohydrodynamics and the physics of 
plasmas is not a sharp one. Nevertheless there are clearly separated 
domains in which the ideas and concepts of only one or the other are 
applicable. One way of seeing the distinction is to look at the way in which 
the relation J = oE is established for a conducting substance. In the 
simple model of Section 7.8 the electrons are imagined to be accelerated by 
the applied fields, but to be altered in direction by collisions, so that their 
motion in the direction of the field is opposed by an effective frictional 
force vm\, where v is the collision frequency. Ohm's law just represents a 

309 



310 Classical Electrodynamics 

balance between the applied force and the frictional drag. When the 
frequency of the applied fields is comparable to v, the electrons have time 
to accelerate and decelerate between collisions. Then inertial effects enter 
and the conductivity becomes complex. Unfortunately at these same 
frequencies the description of collisions in terms of a frictional force tends 
to lose its validity. The whole process becomes more complicated. At 
frequencies well above the collision frequency another thing happens. The 
electrons and ions are accelerated in opposite directions by electric fields 
and tend to separate. Strong electrostatic restoring forces are set up by 
this charge separation. Oscillations occur in the charge density. These 
high-frequency oscillations are called plasma oscillations and are to be 
distinguished from lower-frequency oscillations which involve motion of 
the fluid, but no charge separation. These low-frequency oscillations are 
called magnetohydrodynamic waves. 

In conducting liquids or dense ionized gases the collision frequency is 
sufficiently high even for very good conductors that there is a wide 
frequency range where Ohm's law in its simple form is valid. Under the 
action of applied fields the electrons and ions move in such a way that, 
apart from a high-frequency jitter, there is no separation of charge. 
Electric fields arise from motion of the fluid which causes a current flow, 
or as a result of time-varying magnetic fields or charge distributions 
external to the fluid. The mechanical motion of the system can then be 
described in terms of a- single conducting fluid with the usual hydro- 
dynamic variables of density, velocity, and pressure. At low frequencies 
it is customary to neglect the displacement current in Ampere's law. This 
is then the approximation which is called magnetohydrodynamics. 

In less dense ionized gases the collision frequency is smaller. There 
may still be a low-frequency domain where the magnetohydrodynamic 
equations are applicable to quasi-stationary processes. Frequently astro- 
physical applications fall in this category. At higher frequencies, however, 
the neglect of charge separation and of the displacement current is not allow- 
able. The separate inertial effects of the electrons and ions must be included 
in the description of the motion. This is the domain which we call plasma 
physics. There is here a range of physical conditions where a two-fluid 
model of electrons and ions gives an approximately correct description of 
various phenomena. But for high temperatures and low densities, the 
finite velocity spreads of the particles about their mean values must be 
included. Then the description is made in terms of the Boltzmann 
equation, with or without short-range correlations. We will not attempt 
to go into such details here. At still higher temperatures and lower 
densities, the electrostatic restoring forces become so weak that the length 
scale of charge separation becomes large compared to the size of the 



[Sect. 10.2] Magnetohydrodynamics and Plasma Physics 311 

volume being considered. Then the collective behavior implicit in a fluid 
model is gone completely. We have left a few rapidly moving charged 
particles interacting via Coulomb collisions. A plasma is, by definition, 
an ionized gas in which the length which divides the small-scale individual- 
particle behavior from the large-scale collective behavior is small com- 
pared to the characteristic lengths of interest. This length, called the 
Debye screening radius, will be discussed in Section 10. 10. It is numerically 
equal to 7.91 (T/n) 1A cm, where T is the absolute temperature in degrees 
Kelvin and n is the number of electrons per cubic centimeter. For all but 
the hottest or most tenuous plasmas it is small compared to 1 cm. 



10.2 Magnetohydrodynamic Equations 

We first consider the behavior of an electrically neutral, conducting 
fluid in electromagnetic fields. For simplicity, we assume the fluid to be 
nonpermeable. It is described by a matter density p(x, t), a velocity v(x, /), 
a pressure p(x, t) (taken to be a scalar), and a real conductivity a. The 
hydrodynamic equations are the continuity equation 

^ + V.( P v) = (10.1) 

ot 

and the force equation : 

/>7=-Vp + -(JxB) + F„+ P g (10.2) 

dt c 

In addition to the pressure and magnetic-force terms we have included 
viscous and gravitational forces. For an incompressible fluid the viscous 
force can be written 

F v = r)V 2 v (10.3) 

where r\ is the coefficient of viscosity. It should be emphasized that the 
time derivative of the velocity on the left side of (10.2) is the convective 
derivative, 

l = l + y.V (10.4) 

dt dt 

which gives the total time rate of change of a quantity moving instanta- 
neously with the velocity v. 



312 Classical Electrodynamics 

With the neglect of the displacement current, the electromagnetic fields 
in the fluid are described by 



VxE + i^O 1 
c dt 

VxB = ^J 



(10.5) 



The condition V • J = 0, equivalent to the neglect of displacement 
currents, follows from the second equation in (10.5). The two divergence 
equations have been omitted in (10.5). It follows from Faraday's law that 
(d/dt) V • B = 0, and the requirement V • B = can be imposed as an 
initial condition. With the neglect of the displacement current, it is 
appropriate to ignore Coulomb's law as well. The reason is that the 
electric field is completely determined by the curl equations and Ohm's 
law (see below). If the displacement current is retained in Ampere's law 
and V • E = 4Trp e is taken into account, corrections of only the order of 
(v 2 /c 2 ) result. For normal magnetohydrodynamic problems these are 
completely negligible. 

To complete the specification of dynamical equations we must specify 
the relation between the current density J and the fields E and B. For a 
simple conducting medium of conductivity a, Ohm's law applies, and the 
current density is 

J' = dE! (10.6) 

where J' and E' are measured in the rest frame of the medium. For a 
medium moving with velocity v relative to the laboratory, we must trans- 
form both the current density and the electric field appropriately. The 
transformation of the field is given by equation (6.10). The current density 
in the laboratory is evidently 

J = J' + Pe v (10.7) 

where p e is the electrical charge density. For a one-component conducting 
fluid, p e = 0. Consequently, Ohm's law assumes the form, 

J=(t(e + -xBJ (10.8) 

Sometimes it is possible to assume that the conductivity of the fluid is 
effectively infinite. Then under the action of fields E and B the fluid flows 
in such a way that 

E + - (v x B) = (10.9) 

c 

is satisfied. 



[Sect. 10.3] Magnetohydrodynamics and Plasma Physics 313 

Equations (10.1), (10.2), (10.5), and (10.8), supplemented by an equation 
of state for the fluid, form the equations of magnetohydrodynamics. In 
the next section we will consider some of the simpler aspects of them and 
will elaborate the basic concepts involved. 



10.3 Magnetic Diffusion, Viscosity, and Pressure 

The behavior of a fluid in the presence of electromagnetic fields is 
governed to a large extent by the magnitude of the conductivity. The 
effects are both electromagnetic and mechanical. We first consider the 
electromagnetic effects. We will see that, depending on the conductivity, 
quite different behaviors of the fields occur. The time dependence of the 
magnetic field can be written, using (10.8) to eliminate E, in the form: 

- = Vx(yxB)|-V 2 B (10.10) 

dt Attg 

Here it is assumed that a is constant in space. For a fluid at rest (10.10) 
reduces to the diffusion equation 

?5 = _£!_V 2 B (10.11) 

dt Attg 

This means that an initial configuration of magnetic field will decay away 
in a diffusion time 

r = *=£ (10.12) 

where L is a length characteristic of the spatial variation of B. The time t 
is of the order of 1 sec for a copper sphere of 1 cm radius, of the order of 10 4 
years for the molten core of the earth, and of the order of 10 10 years for a 
typical magnetic field in the sun. 

For times short compared to the diffusion time t (or, in other words, 
when the conductivity is so large that the second term in (10.10) can be 
neglected) the temporal behavior of the magnetic field is given by 

3D 

u — = V x (v x B) (10.13) 

dt 

From (6.5) it can be shown that this is equivalent to the statement that the 
magnetic flux through any loop moving with the local fluid velocity is 
constant in time. We say that the lines of force are frozen into the fluid 
and are carried along with it. Since the conductivity is effectively infinite, 



314 Classical Electrodynamics 

the velocity w of the lines of force (defined to be perpendicular to B) is 
given by (10.9): 

w = c (ExB) M) 

B 2 

This so-called "E X B drift" of both fluid and lines of force can be under- 
stood in terms of individual particle orbits of the electrons and ions in 
crossed electric and magnetic fields (see Section 12.8). 

A useful parameter to distinguish between situations in which diffusion 
of the field lines relative to the fluid occurs and those in which the lines of 
force are frozen in is the magnetic Reynolds number R M . If V is a velocity 
typical of the problem and L is a corresponding length, then the magnetic 
Reynolds number is defined as 

R M = -^ (10.15) 

where r is the diffusion time (10.12). Transport of the lines of force with 
the fluid dominates over diffusion if R M > 1 . For liquids like mercury or 
sodium in the laboratory R M < 1, except for very high velocities. But in 
geophysical and astrophysical applications R M can be very large compared 
to unity. 

The mechanical behavior of the system can be studied with the force 
equation (10.2). Substituting for J from (10.8), we find 

P7 = F-4 2 (vi -w) (10.16) 

dt c 2 

where F is the sum of all the nonelectromagnetic forces, and y is the 
component of velocity perpendicular to B. From (10.16) it is apparent 
that flow parallel to B is governed by the nonelectromagnetic forces alone. 
The velocity of flow of the fluid perpendicular to B, on the other hand, 
decays from some initially arbitrary value in a time of the order of 



pc 2 

T = 



—. ; (10-17) 

oB 2 



to a value 

v i = W + -^ i F 1 (10.18) 

CfB 

In the limit of infinite conductivity this result reduces to that of (10.14), 
as expected. The term proportional to B 2 in (10.16) is an effective viscous 
or frictional force which tends to prevent flow of the fluid perpendicular to 
the lines of magnetic force. Sometimes it is described as a magnetic 
viscosity. If ordinary viscosity, here lumped into F, is comparable to the 



[Sect. 10.3] Magnetohydrodynamics and Plasma Physics 315 

magnetic viscosity, then the decay time r is shortened by an obvious factor 
involving the ratio of the two viscosities. 

The above considerations have shown that if the conductivity is large 
the lines of force are frozen into the fluid and move along with it. Any 
departure from that state decays rapidly away. In considering the 
mechanical or electromagnetic effects we treated the opposite quantities as 
given, but the equations are, of course, coupled. In the limit of very large 
conductivity it is convenient to relate the current density J in the force 
equation to the magnetic induction B via Ampere's law and to use the 
infinite conductivity expression (10.9) to eliminate E from Faraday's law 
to yield (10.13). The magnetic force term in (10.2) can now be written 

- (J x B) = - — B x (V x B) (10.19) 

C 477 

With the vector identity 

|V(B • B) = (B • V)B + B x (V x B) (10.20) 

Equation (10.19) can be transformed into 

- (J x B) = -V (f) + 1 (B • V)B (10.21) 

This equation shows that the magnetic force is equivalent to a magnetic 
hydrostatic pressure, 2 

Pm = ^- (10-22) 

07T 

plus a term which can be thought of as an additional tension along the 
lines of force. The result (10.21) can also be derived frorh the Maxwell 
stress tensor (see Section 6.9). 

If we neglect viscous effects and assume that the gravitational force is 
derivable from a potential g = — Vip, the force equation (10.2) takes the form 

/»7= - V(p + p M + PV>) + 7" (B • V)B (10.23) 

dt 4tt 

In some simple geometrical situations, such as B having only one com- 
ponent, the additional tension vanishes. Then the static properties of the 

fluid are described by „«.» 

J p + p M + py> = constant (10.24) 

This shows that, apart from gravitational effects, any change in mechanical 
pressure must be balanced by an opposite change in magnetic pressure. If 
the fluid is to be confined within a certain region so that p falls rapidly to 
zero outside that region, the magnetic pressure must rise equally rapidly 
in order to confine the fluid. This is the principle of the pinch effect 
discussed in Section 10.5. 



316 



Classical Electrodynamics 



10.4 Magnetohydrodynamic Flow between Boundaries with Crossed 
Electric and Magnetic Fields 



To illustrate the competition between freezing in of lines of force and 
diffusion through them and between the E x B drift and behavior imposed 
by boundary conditions, we consider the simple example of an incom- 
pressible, but viscous, conducting fluid flowing in the x direction between 
two nonconducting boundary surfaces at z = and z = a, as shown in 
Fig. 10.1. The surfaces move with velocities V x and V 2 , respectively, in 
the x direction. A uniform magnetic field B acts in the z-direction. The 
system is infinite in the x and y directions. We will look for a steady-state 
solution for flow in the x direction in which the various quantities depend 
only upon z. 

If the fields do not vary in time, it is clear from Maxwell's equations 
(10.5) that any electric field present must be an electrostatic field derivable 
from a potential and determined solely by the boundary conditions, i.e. 
an arbitrary external field. Expression (10.14) for the velocity of the lines 
of force when a is infinite implies that there is an electric field in the y 
direction. If we assume that to be the only component of E, then it must 
be a constant, E . Because the moving fluid will tend to carry the lines of 
force with it, we expect an x component BJz) of magnetic induction, as 
well as the z component B . 

The continuity equation (10. 1) reduces to V • v = for an incompressible 
fluid. This is satisfied identically by a velocity in the x direction which 
depends only on z. The force equation, neglecting gravity, has the steady- 
state form: 



Vp = -(J x B) + rjV 2 v 
c 



(10.25) 




Fig. 10.1 Flow of viscous con- 
ducting fluid in a magnetic field 
between two plane surfaces 
moving with different velocities. 



[Sect. 10.4] Magnetohydrodynamics and Plasma Physics 

The only component of J that is nonvanishing is J y (z) : 



J y (z) = a 



E -- B v(z) 
c 



317 



(10.26) 



where v is the x component of velocity. When we write out the three 
component equations in (10.25), we find 



dp aBJ^ B \ d 2 v 

ox c \ c I oz 

dp 

dy 



^ = 



dp 

dz 



-*(*.-*.) 



(10.27) 



The magnetic force in the z direction is just balanced by the pressure 
gradient. If we assume no pressure gradient in the x direction, the first of 
these equations can be written : 

d 2 v 



dz 2 



where 



Tif— m 



2„2\V£ 



M 



\ TjC 2 I 



(10.29) 



is called the Hartmann number. From (10.17) M 2 can be seen to be the 
ratio of magnetic to normal viscosity. The solution to (10.28), subject to 
the boundary conditions v(0) = V x and v(a) = V 2 , is readily found to be 



v(z) = - 



V x 



sinh M 



sinh 





In the limit B n 



(10.30) 

sinh M 
0, M — *■ 0, we obtain the standard laminar-flow result 

z 



v(z)= V X + -{V % 
a 



V,) 



(10.31) 



In the other limit of M > 1 we expect the magnetic viscosity to dominate 
and the flow to be determined almost entirely by the E x B drift. If we 
approximate v(z) for z < a and M > 1, we obtain 

V(Z) - ^2 + ( Vl _ £Eo\ e -Mz/a (1Q 32) 

B \ B Q J 



318 



Classical Electrodynamics 




m:»i 



Fig. 10.2 Velocity profiles for 
large and small Hartmann 
numbers M. For M -*■ 0, lami- 
nar flow occurs. ForMg> l,the 
flow is given by the E x B drift 
velocity, except in the immediate 
neighborhood of the boundaries. 



This shows that, while v(z) = V x exactly at the surface, there is a rapid 
transition in a distance of order (a/M) to the E x B drift value (cEJBq). 
Near z = a, (10.32) is changed by replacing V 1 by V 2 and z by (a — z). The 
velocity profile in the two limits (10.31) and (10.32) is shown in Fig. 10.2. 
The magnetic field B x (z) is determined by the equation 



dB x _ 4tt _ Arra . 
oz c c 



o-- B o y ) 

C I 



(10.33) 



The boundary conditions on B x at z = and z — a are indeterminate 
unless we know the detailed history of how the steady state was created or 
can use some symmetry argument. All we know is that the difference in 
B x is related to the total current flowing in the y direction per unit length 
in the x direction : 



B x (a) - B x (0) 



c Jo 



(z) dz 



(10.34) 



This indeterminacy stems from the one-dimensional nature of the problem. 
For simplicity we will calculate the magnetic field only for the case when 
the total current in the y direction is zero. * Then we can assume that B x 
vanishes at z = and z — a. Using (10.30) for the velocity in (10.33), it 
is easy to show that then 



BJz) = B 



/ 47rora 2 



)(*=*)] 



cosh cosh 

2 



(M Mz\ 



M sinh — 

2 



(10.35) 



* This requirement means that cE /B = ^{V x + V 2 ). 



[Sect. 10.4] 



Magnetohydrodynamics and Plasma Physics 



319 



The dimensionless coefficient in square brackets in (10.35) may be identified 
as the magnetic Reynolds number (10.15), since (V 2 — V^/2 is a typical 
velocity in the problem and a is a typical length. In the two limits M < 1 
and M > 1, (10.35) reduces to 



BM ~ R M B, 



a\ al 

l[l-(e 

.ML \ 



Mz 



+ e 



-m(2=£) 



)] 



for M < 1 



for M > 1 



(10.36) 



Figure 10.3 shows the behavior of the lines of force in the two limiting 
cases. Only for large R M is there appreciable transport of the lines of force. 
And for a given R M , the transport is less the larger the Hartmann number. 
For liquid mercury at room temperature the relevant physical constants 
are 

a = 9.4 x 10 15 sec -1 

r\ — 1.5 X 10 -2 poise 

p = 13.5 gm/cm 3 

The diffusion time (10.12) is t = 1.31 x 10" 4 [L (cm)] 2 sec. The Hartmann 
number (10.29) is M = 2.64 X lO" 2 ^ (gauss) a (cm). With L ~ a ~ 1 cm, 
this gives a magnetic Reynolds number R M ~ IGr^V. Consequently 
unless the flow velocity is very large, there is no significant transport of 
lines of force for laboratory experiments with mercury. On the other hand, 
if the magnetic induction B is of the order of 10 4 gauss, then M ~ 250 and 
the velocity flow is almost completely specified by the E x B drift (10.14). 




^V 2 >Vi 




M«\ 



M»l 



(a) 



(b) 



Fig. 10.3 (a) Axial component of magnetic induction between the boundary surfaces 
for large and small Hartmann numbers. (6) Transport of lines of magnetic induction in 

direction of flow. 



320 Classical Electrodynamics 

In geomagnetic problems with the earth's core and in astrophysical 
problems the parameters (e.g., the length scale) are such that R M > 1 
occurs often and transport of the lines of force becomes very important. 

10.5 Pinch Effect 

The confinement of a plasma or conducting fluid by self-magnetic fields 
is of considerable interest in thermonuclear research, as well as in other 
applications. To illustrate the principles we consider an infinite cylinder 
of conducting fluid with an axial current density J z = J(r) and a resulting 
azimuthal magnetic induction B+ = B{r). For simplicity, the current 
density, magnetic field, pressure, etc., are assumed to depend only on the 
distance r from the cylinder axis, and viscous and gravitational effects are 
neglected. We first ask whether a steady-state condition can exist in which 
the material is mainly confined within a certain radius r = R by the 
action of its own magnetic induction. For a steady state with v = the 
equation of motion (10.23) of the fluid reduces to 

n dp d(B 2 \ B 2 

Ampere's law in integral form relates B{r) to the current enclosed: 

4tt f r 
B(r) = — rJ(r) dr (10.38) 

cr Jo 

A number of results can be obtained without specifying the form of J(r), 
aside from physical limitations of finiteness, etc. From Ampere's law it is 
evident that, if the fluid lies almost entirely inside r = R, then the mag- 
netic induction outside the fluid is 

B(r) = — (10.39) 

cr 
where 



-I 



R 

27rrJ(r) dr 



is the total current flowing in the cylinder. Equation (10.37) can be 
written as 

dp Id, 2r ,<K 

Jr = ~^7r (rB) (la40 > 

with the solution : 

P(r) = Po-j~\ r ~ (r 2 5 2 ) dr (10.41) 

8tt- Jo r 2 dr 



[Sect. 10.5] Magnetohydrodynamics and Plasma Physics 321 

Here p is the pressure of the fluid at r = 0. If the matter is confined to 
r < R, the pressure drops to zero at r = R. Consequently the axial 
pressure p is given by 

p° = v- r L A^ B2)dr (i °- 42) 

Stt Jo r* dr 

The upper limit of integration can be replaced by infinity, since the inte- 
grand vanishes for r > R, as can be seen from (10.39). With this expression 
(10.42) forp , (10.41) can be written as 

p(r) = f \ R \l(r*J?)dr (10.43) 

877 Jr r* dr 

The average pressure inside the cylinder can be related to the total 
current / and radius R without specifying the detailed radial behavior. 
Thus 

<P>=^ 2 \ B rp(r)dr (10.44) 

ttR 2 Jo 

Integration by parts and use of (10.40) gives 

<P> = — ^— (10.45) 

as the relation between average pressure, total current, and radius of the 
cylinder of fluid or plasma confined by its own magnetic field. Note that 
the average pressure of the matter is equal to the magnetic pressure (B 2 I$tt) 
at the surface of the cylinder. In thermonuclear work, hot plasmas with 
temperatures of the order of 10 8 °K (kT~ lOkev) and densities of the 
order of 10 15 particles/cm 3 are envisioned. These conditions correspond 
to a pressure of approximately 10 15 x I0 8 k ^ 1.4 x 10 7 dynes/cm 2 , or 14 
atmospheres. A magnetic induction of approximately 19 kilogauss at the 
surface, corresponding to a current of 9 x 10 4 /? (cm) amperes, is necessary 
for confinement. This shows that extremely high currents are needed to 
confine very hot plasmas. 

So far the radial behavior of the system has not been discussed. Two 
simple examples will serve to illustrate the possibilities. One is that the 
current density J(r) is constant for r < R. Then B(r) = (llr/cR 2 ) for 
r < R. Equation (10.43) then yields a parabolic dependence for pressure 
versus radius : 

^-^f 1 -?) (ia46) 

The axial pressure p is then twice the average pressure (p). The radial 
dependences of the various quantities are sketched in Fig. 10.4. 



322 



Classical Electrodynamics 




Fig. 10.4 Variation of azimuthal 

magnetic induction and pressure with 

radius in a cylindrical plasma column 

with a uniform current density /. 



The other model has the current density confined to a very thin layer on 
the surface, as is appropriate for a highly conducting fluid or plasma. The 
magnetic induction is given by (10.39) for r > R, but vanishes inside the 
cylinder. Then the pressure p is constant inside the cylinder and equal to 
the value (10.45). This is sketched in Fig. 10.5. 



10.6 Dynamic Model of the Pinch Effect 

The simple considerations of the previous section are valid for a static 
or quasi-static situation. In actual practice with plasmas, such circum- 
stances do not arise. Generally, at some time early in the history of 
current flow down the plasma the pressure/? is much too small to resist the 
magnetic pressure outside. Consequently the radius of the cylinder of 
plasma is forced inwards; the plasma column is pinched. This has the 
desirable consequence that the plasma is pulled away from its confining 
walls. If the pinched configuration were stable for a sufficiently long time, 
it would be possible to heat the plasma to very high temperatures without 
burning up the walls of the confining vessel. 




Fig. 10.5. Variation of azi- 
muthal magnetic induction and 
pressure in a cylindrical plasma 
column with a surface-current 
density. 



[Sect. 10.6] 



Magnetohydrodynamics and Plasma Physics 



323 



A simple model, first discussed by M. Rosenbluth, exhibits the essential 
dynamical features. Suppose that a plasma is created in a hollow con- 
ducting cylinder of radius Rq and length L. A voltage difference V is 
applied between the ends of the cylinder so that a current J flows in the 
plasma. This produces an azimuthal magnetic induction B^ which causes 
the plasma to pinch inwards. The radius of the plasma column at time 
t > is R(t). The conductivity of the plasma is taken to be virtually 
infinite. Then the current all flows on the surface, and the magnetic 
induction 

B+ = - (10.47) 

cr 

exists only between r = R(t) and r = Rq. Because of the assumption of 
infinite conductivity the electric field at the plasma surface, in the moving 
frame of reference in which the interface is at rest, vanishes : 



E' = E + - xB = 



(10.48) 



If we now apply Faraday's law of induction to the dotted loop shown in 
Fig. 10.6, the inner arm of which is moving inwards with the interface, we 
find that the only contribution to the line integral of E comes from the 
side of the loop in the conducting wall. Thus 



__!i (\dr--l±(im*9) (10.49) 

cdtJmt)* c 2 dt\ R/ v ' 



This is the standard inductive relation between current, voltage, and 
dimensions (inductance). The integral of this equation is 



ta (£)-W>* 



(10.50) 



where E f(f) = V/L is the applied electric field. The function fit) is 
assumed known and is normalized so that E is the peak value of applied 



$ = o *= v 




Fig. 10.6 Plasma column inside a hollow, 
cylindrical conductor. 



■< L *► 



324 Classical Electrodynamics 

field. In order to proceed further we must relate the current I'm a dynamic 
way to the behavior of the plasma radius R. 

The desired dynamical connection between I and R is essentially the 
momentum-balance equation, or Newton's second law. Some assumption 
about the plasma must be made. If the mean free path for collisions is 
short compared to the radius, the dynamic behavior is characteristic of 
hydrodynamic shock waves. But for a hot, tenuous plasma the mean free 
path is comparable to, or larger than, the radius. Then a model with 
particles moving freely inside the plasma is more appropriate. If the 
velocity A of the plasma surface is large compared to thermal speeds, each 
particle approaches the interface with a velocity A in the frame of reference 
in which the interface is at rest. As the particle penetrates into the outer 
region, it starts feeling the magnetic induction, is turned around, and leaves 
the surface with velocity A. Consequently each particle colliding with the 
plasma surface receives a momentum transfer 2MA. The number colliding 
with unit area of the surface per unit time is NR, where N is the initial 
number of particles per unit volume. Therefore the rate of transfer of 
momentum per unit area (i.e., pressure) is 

p = 2NMR* = 2 P R 2 (10.51) 

where p is the initial mass density. At the surface of the plasma there is a 
magnetic pressure CB 2 /8tt) due to the discontinuity in magnetic induction 
from zero inside to the value B just outside. These pressures must balance. 
Consequently, using (10.47), we find that the current is related to the 
velocity by : 

7 2 = 4"PC 2 R 2 (^f (10.52) 

Equation (10.52) depends on a rather simplified model of the mechanical- 
momentum transfer rate in which each particle collides only once with the 
interface. In fact, the velocity of the interface increases with time so that 
the surface catches up with particles which were reflected earlier and hits 
them again and again. This effect can be approximated by the "snow- 
plow" model in which the interface is imagined to carry along with it all 
the material which it hits as it moves in. Then the magnetic pressure and 
rate of change of momentum are related by 



d 
dtl 



M(R)A 



d2 

■2ttR— (10.53) 

07T 



where M(R) is the mass carried along by the snowplow: 

M(R) = tt P {R* - i?2) ( 10 .54) 



[Sect. 10.6] 



Magnetohydrodynamics and Plasma Physics 



325 



This leads to the relation 



/2 = -npc^R 



dtl 



(Ro 2 -R 2 )f A 



(10.55) 



between current and radius. In the initial stages when R < R the snow- 
plow model and free-particle model give the same relation between current 
and radius to within a factor of 2*, and do not differ by an order of 
magnitude even at later times. 

The equation of motion for R(t) is obtained by substituting P from 
either (10.52) or (10.55) into the inductive relation (10.50). Choosing the 
free-particle model as an illustration, we obtain 

2R In (- )— = - -£&= [fit') df (10.56) 

\ R 1 dt y/Anp Jo 

where the signs of the square root have been taken to give R < 0. Without 
knowledge of f(t) we cannot solve this equation. Nevertheless, some 
idea of the solution can be obtained by introducing the dimensionless 
variables : 

Wl " 
\ 47773/ R 



_ R_ 
X ~ R. 



Then (10.56) becomes 



2x\nx— = [ fir') dr' 
dr Jo 

For the snowplow model the equivalent equation is 



4 

drl 



<*-< 



(10.57) 



(10.58) 



(10.59) 



M 



(10.60) 



z(ln zf 

Without solving these equations it is evident that x changes significantly 
in times such that t ~ 1. This means that the scaling law for the radial 
velocity of the pinch is 

\Airp J 

This result emerges whatever dynamic model is used, including a hydro- 
dynamic one. Typical experimental conditions for a fast pinch in small- 
scale hydrogen or deuterium plasmas involve applied electric fields of the 

* The factor of 2 comes from the fact that in the one case the particles are elastically 
reflected and suffer a velocity change of 2R, while in the other the particles collide 
inelastically with the interface and receive a velocity change of R. 



326 



Classical Electrodynamics 




Fig. 10.7 Radius of plasma 
column as a function of time 
after initiation of current flow. 
The characteristic velocity of 
pinching is given by (10.60). 



order of 10 3 volts/cm and initial densities of the order of 10 -8 gm/cm 3 
(~ 3 X 10 15 deuterons/cm 3 ). Then v is of the order of 10 7 cm/sec. The 
current flowing is, according to (10.52) or (10.55), 



v n \ dr) 



(10.61) 



where F is a dimensionless function of the order of unity. For a tube 
radius of 10 cm and the conditions described, the current /is measured in 
units of 10 5 or 10 6 amperes. 

The discussion of the pinching action presented so far is obviously valid 
only for short times after the initiation of current flow. The simplified 
models indicate that in a time of the order of R /v the radius of the plasma 
column goes to zero. It is clear, however, that before that will occur (even 
approximately) the behavior will be modified. In the hydrodynamic limit, 
the radial shock waves caused by the pinch will be reflected off the axis 
and move outwards, striking the interface and retarding its inward motion 
or even reversing it. This phenomenon is known as bouncing. It is 
evidently present also in the free-particle model. Consequently the general 
behavior of radius R as a function of time is expected to be as shown in 
Fig. 10.7. Although no proper analysis has been made of the subsequent 
bounces, it is conjectured that there is an approach to a steady state at 
some radius less than R . 



10.7 Instabilities in a Pinched-PIasma Column 



In the laboratory long-lived pinched plasmas are extremely difficult to 
produce. The dynamic behavior of the previous section is found to be 
followed at least qualitatively for times up to around the first bounce. 
But then the plasma column is observed to break up rapidly. The reason 
for the disintegration of the column is the growth of instabilities. The 



[Sect. 10.7] 



Magnetohydrodynamics and Plasma Physics 



327 




Fig. 10.8 (a) Kink instability. 
(6) Sausage or neck instability. 



(b) 



column is unstable against various departures from cylindrical geometry. 
Small distortions are amplified rapidly and destroy the column in a very 
short time. The detailed analysis of instabilities is sufficiently complex 
that we will attempt only qualitative arguments. Two of the simpler 
unstable distortions will be described. 

The first is the kink instability, shown in Fig. 10.8a. The lines of azimu- 
thal magnetic induction near the column are bunched together above, and 
separated below, the column by the distortion downwards. Thus the 
magnetic pressure changes are in such a direction as to increase the 
distortion. The distortion is unstable. 

The second type of distortion is called a sausage or neck instability, 
shown in Fig. 10.86. In the neighborhood of the constriction the azi- 
muthal induction increases, causing a greater inwards pressure at the neck 
than elsewhere. This serves to enhance the existing distortion. 

Both types of instability are hindered by axial magnetic fields within 
the plasma column. For the sausage distortion the lines of axial induction 
are compressed by the constriction, causing an increased pressure inside 
to oppose the increased pressure of the azimuthal field, as indicated 
schematically in Fig. 10.9. It is easy to see that the fractional changes in 



Fig. 10.9 Hindering neck instability with 

outward pressure of trapped axial magnetic 

fields. 




328 



Classical Electrodynamics 




Fig. 10.10 Hindering kink instability with tension of trapped axial fields. 



the two magnetic pressures, assuming a sharp boundary to the plasma, 

are A ~ A 

Ap^ = 2x Ap z _ 4x 
P4> ~ R ' p z R 
where x is the small inwards displacement. Consequently, if 



(10.62) 



B? > *V 



(10.63) 



the column is stable against sausage distortions. 

For kinks the axial magnetic field lines are stretched, rather than com- 
pressed laterally together. But the result is the same ; namely the increased 
tension in the field lines inside opposes the external forces and tends to 
stabilize the column. It is evident from Fig. 10.10 that a short- wavelength 
kink of a given lateral displacement will cause the lines of force to stretch 
relatively more than a long-wavelength kink. Consequently, for a given 
ratio of internal axial field to external azimuthal field, there will be a 
tendency to stabilize short-wavelength kinks, but not very long-wavelength 
ones. If the fields are approximately equal, analysis shows that if the wave- 
length of the kink X < 14 R the disturbance is stabilized. 

For longer-wavelength kinks stabilization can be achieved by the action 
of the outer conductor, provided the plasma radius is not too small 
compared to the radius of the conductor. The azimuthal field lines are 
trapped between the conductor and the plasma boundary, as shown in 
Fig. 10.11. If the plasma column moves too close to the walls, the lines 
of force are crowded together between it and the walls, causing an in- 
creased magnetic pressure and restoring force. 



Fig. 10.11 Stabilization of 

long-wavelength kinks with 

outer conductor. 




[Sect. 10.8] Magnetohydrodynamics and Plasma Physics 329 

It is clear qualitatively that it must be possible, by a combination of 
trapped axial field and conducting walls, to create a stable configuration, 
at least in the approximation of a highly conducting plasma with a sharp 
boundary. Detailed analysis* confirms this qualitative conclusion and 
sets limits on the quantities involved. It is important to have as little axial 
field outside the plasma as possible and to keep the plasma radius of the 
order of one-half or one-third of the cylinder radius. If the axial field 
outside the plasma is too large, the combined B z and B+ cause helical 
instabilities that are troublesome in toroidal geometries. If, however, the 
axial field outside the plasma is made very large, the pitch of the helix 
becomes so great that there is much less than one turn of the helix in a 
plasma column of finite length. Then it turns out that there is the possi- 
bility of stability again. Stabilization by means of a strong axial field 
produced by currents external to the plasma is the basis of some fusion 
devices, e.g., the Stellarator. 

The idealized situation of a sharp plasma boundary is difficult to create 
experimentally, and even when created is destroyed by diffusion of the 
plasma through the lines of force in times of the order of AttgR 2 \c 2 (see 
Section 10.3). For a hydrogen plasma of 1 ev energy per particle this time 
is of the order of 10 -4 sec for R ~ 10 cm, while for a 10 kev plasma it is 
of the order of 10 2 sec. Clearly the thermonuclear experimenter must try 
to create initially as hot a plasma as possible in order to make the initial 
diffusion time long enough to allow further heating. 



10.8 Magnetohydrodynamic Waves 

In ordinary hydrodynamics the only type of small-amplitude wave 
motion possible is that of longitudinal, compressional (sound) waves. 
These propagate with a velocity s related to the derivative of pressure with 
respect to density at constant entropy: 



Wo 



(10.64) 
dp/ 

If the adiabatic law p = Kp y is assumed, s 2 = yp lp , where y is the ratio 
of specific heats. In magnetohydrodynamics another type of wave motion 
is possible. It is associated with the transverse motion of lines of magnetic 
force. The tension in the lines of force tends to restore them to straight- 
line form, thereby causing a transverse oscillation. By analogy with 

* V. D. Shafranov, Atomnaya Energ. I, 5, 38 (1956); R. J. Tayler, Proc. Phys. Soc. 
{London), B70, 1049 (1957); M. Rosenbluth, Los Alamos Report L/l-2030 (1956). See 
also Proceedings of the Second International Conference on Peaceful Uses of Atomic 
Energy, Vol. 31 (1958), papers by Braginsky and Shafranov (p. 43) and Tayler (p. 160). 



330 



Classical Electrodynamics 



ordinary sound waves whose velocity squared is of the order of the hydro- 
static pressure divided by the density, we expect that these magnetohydro- 
dynamic waves, called Alfven waves, will have a velocity 

v A ~(p-\ (10.65) 

where B 2 I&it is the magnetic pressure. 

To examine the wave motion of a conducting fluid in the presence of a 
magnetic field, we consider a compressible, nonviscous, perfectly con- 
ducting fluid in a magnetic field in the absence of gravitational forces. The 
appropriate equations governing its behavior are : 



dt 



(py) = o 



p^+/>(v.V)v= -v P 
at 

— = V x (v x B) 

dt 



— B x (V x B) 

4tt 



(10.66) 



These must be supplemented by an equation of state relating the pressure 
to the density. We assume that the equilibrium velocity is zero, but that 
there exists a spatially uniform, static, magnetic induction B throughout 
the uniform fluid of constant density p . Then we imagine small-amplitude^ 
departures from the equilibrium values : 

B = B o + B 1 (x,0" 

P = Po + fti(x, ► 00.67) 

v = Vi(x, J 

If equations (10.66) are linearized in the small quantities, then they 
become : 

dpi 
Bt 



+ / , V.v 1 = 



Po ^ + s*V Pl + ^x(VxB 1 ) = 
dt Att 



dBi 

dt 



- V x ( Vl x B ) = 



(10.68) 



where s 2 is the square of the sound velocity (10.64). These equations can 
be combined to yield an equation for v x alone : 

?^ _ S 2 V (V . Vl ) + v^ x V x [V x ( Vl x yj] = (10.69) 
dt 2 



[Sect. 10.8] Magnetohydrodynamics and Plasma Physics 331 

where we have introduced a vectorial Alfven velocity: 

y A = -^= (10.70) 

The wave equation (10.69) for v x is somewhat involved, but it allows 
simple solutions for waves propagating parallel or perpendicular to the 
magnetic field direction.* With v 1 (x, t) a plane wave with wave vector k 
and frequency co : 

Vl (x, /) = Vie*-*"** (10.71) 

equation (10.69) becomes: 

-a> 2 v 1 + (.s 2 + VX k 'Vi)k 

+ v A • k[(v^ • k)v x - {y A • v x )k - (k • Vl )vJ = (10.72) 

If k is perpendicular to v A the last term vanishes. Then the solution for v x 
is a longitudinal magnetosonic wave with a phase velocity : 



"long = V* 2 + v A * (10.73) 

Note that this wave propagates with a velocity which depends on the sum 
of hydrostatic and magnetic pressures, apart from factors of the order of 
unity. If k is parallel to \ A , (10.72) reduces to 



(5-) 



(*V - <>vi + [f- 2 - 1 J k\y A • Vl )v, = (10.74) 

There are two types of wave motion possible in this case. There is an 
ordinary longitudinal wave (v x parallel to k and v^) with phase velocity 
equal to the sound velocity s. But there is also a transverse wave (v x • v A = 
0) with a phase velocity equal to the Alfven velocity v A . This Alfven wave 
is a purely magnetohydrodynamic phenomenon which depends only on 
the magnetic field (tension) and the density (inertia). 

For mercury at room temperature the Alfven velocity is [B (gauss)/ 13.1]) 
cm/sec, compared with sound speed of 1.45 x 10 5 cm/sec. At all labora- 
tory field strengths the Alfven velocity is much less than the speed of 
sound. In astrophysical problems, on the other hand, the Alfven velocity 
can become very large because of the much smaller densities. In the sun's 
photosphere, for example, the density is of the order of 10~ 7 gm/cm 3 
(~6 x 10 16 hydrogen atoms/cm 3 ) so that v A ^ 10 3 B cm/sec. Solar 
magnetic fields appear to be of the order of 1 or 2 gauss at the surface, with 
much larger values around sunspots. For comparison, the velocity of 
sound is of the order of 10 6 cm/sec in both the photosphere and the 
chromosphere. 

* The determination of the characteristics of the waves for arbitrary direction of 
propagation is left to Problem 10.3. 



332 



Classical Electrodynamics 



,* n mm 



m m M'*A 



/^ M M M M An 



A h M A II >lM 



MM A A 



An A m 




(a) (6) 

Fig. 10.12 Magnetohydrodynamic waves. 

The magnetic fields of these different waves can be found from the third 
equation in (10.68): 

r k 



B x = 



-«A 


for klB 





for the longitudinal k || B 


5 V 1 

ft) 


for the transverse k || B 



(10.75) 



The magnetosonic wave moving perpendicular to B causes compressions 
and rarefactions in the lines of force without changing their direction, as 
indicated in Fig. 10. 12a. The Alfven wave parallel to B causes the lines of 
force to oscillate back and forth laterally (Fig. 10.126). In either case the 
lines of force are "frozen in" and move with the fluid. 

If the conductivity of the fluid is not infinite or viscous effects are present, 
we anticipate dissipative losses and a consequent damping of oscillations. 
The second and third equations in (10.68) are modified by additional terms : 



Po ?ll = _*» V Pl - 5-° x (V x B x ) + r, V\ 

Ot 47T 

^i = V x( Vl xB ) + /-V 2 B 1 
dt 4tto 



(10.76) 



where r\ is the viscosity* and a is the conductivity. Since both additions 
cause dispersion in the phase velocity, their effects are most easily seen 
when a plane wave solution is being sought. For plane waves it is evident 

* Use of the simple viscous force (10.3) is not really allowed for a compressible fluid. 
But it can be expected to give the correct qualitative behavior. 



[Sect. 10.8] Magnetohydrodynamics and Plasma Physics 333 

that these equations are equivalent to 



dy 1 1 

dt 



\ Po (o/ 

dB t 1 






-s*V Pl -^ x(V xBO 



V x ( Vl x B ) 



(10.77) 



Consequently equation (10.72) relating k and a> is modified by (a) multi- 

/ c z k 2 \ 
plying s 2 and to 2 by the factor I 1 + / 1 , and (b) multiplying a> 2 by 

/ k 2\ \ AttOW} 

the factor 1 + i — . 

\ p Q M/ 

For the important case of the Alfven wave parallel to the field, the 
relation between a> and k becomes 

fcV = ^ 2 (l + * —) (l + * —) (10.78) 

\ AttO(jo' \ PqO)' 

If the resistive and viscous correction terms are small, the wave number is 
approximately 

k ~«L +t MjL + 3\ (10.79) 

v A 2vJ\4na Po / 

This shows that the attenuation increases rapidly with frequency (or wave 
number), but decreases with increasing magnetic field strength. In terms 
of the diffusion time t of Section 10.3, the imaginary part of the wave 
number shows that, apart from viscosity effects, the wave travels for a time 
t before falling to 1/e of its original intensity, where the length parameter 
in t (10.12) is the wavelength of oscillation. For the opposite extreme in 
which the resistive and/or viscous terms dominate, the wave number is 
given by the vanishing of the two factors on the right-hand side of (10.78). 
Thus k has equal real and imaginary parts and the wave is damped out 
rapidly, independent of the magnitude of the magnetic field. 

The considerations of magnetohydrodynamic waves given above are 
valid only at comparatively low frequencies, since the displacement 
current was ignored in Ampere's law. It is evident that, if the frequency is 
high enough, the behavior of the fields must go over into the "ionospheric" 
behavior described in Section 7.9, where charge-separation effects play an 
important role. But even when charge-separation effects are neglected in 
the magnetohydrodynamic description, the displacement current modifies 
the propagation of the Alfven and magnetosonic waves. The form of 



334 Classical Electrodynamics 

Ampere's law, including the displacement current, is : 

VxB = -J--,-(vxB) (10.80) 

c c 2 dt 

where we have used the infinite conductivity approximation (10.9) in 
eliminating the electric field E. Thus the current to be inserted into the 
force equation for fluid motion is now 



J = — 

4-7T 



VxB + - 2 |(vxB) 
c 2 dt 



(10.81) 



In the linearized set of equations (10.68) the second one is then generalized 
to read: 



Po 



£ + M§"*)]— ' Vft -5" <y,, * ) (1082) 



'Ldt 
This means that the wave equation for v x is altered to the form: 



\(i + v 4)-H(^^) 



dfL x \ c 2 / c 



-s 2 V(V. Vl ) 

+ T i xVxVx(T 1 xyJ = (10.83) 



Inspection shows that for y 1 parallel to v A (i.e., B ) there is no change from 
before. But for transverse ^ (either magnetosonic with k perpendicular 
to B , or Alfven waves with k parallel to B ) the square of the frequency is 
multiplied by a factor [1 + (v A 2 lc 2 )]. Thus the phase velocity of Alfven 
waves becomes 

u A = , CVa (10.84) 

Vc 2 + vj 

In the usual limit where v A < c, the velocity is approximately equal to v A ; 
the displacement current is unimportant. But, if v A > c, then the phase 
velocity is equal to the velocity of light. From the point of view of electro- 
magnetic waves, the transverse Alfven wave can be thought of as a wave 
in a medium with an index of refraction given by 



Thus 



u A = - (10.85) 

n 

n 2 = 1 + — = 1 + ^4- < ia86 > 

Caution must be urged in using this index of refraction for the propagation 
of electromagnetic waves in a plasma. It is valid only at frequencies where 
charge-separation effects are unimportant. 



[Sect. 10.9] Magnetohydrodynamics and Plasma Physics 335 



10.9 High-Frequency Plasma Oscillations 

The magnetohydrodynamic approximation considered in the previous 
sections is based on the concept of a single-component, electrically neutral 
fluid with a scalar conductivity a to describe its interaction with the 
electromagnetic field. As discussed in the introduction to this chapter a 
conducting fluid or plasma is, however, a multicomponent fluid with 
electrons and one or more types of ions present. At low frequencies or 
long wavelengths the description in terms of a single fluid is valid because 
the collision frequency v is large enough (and the mean free path short 
enough) that the electrons and ions always maintain local electrical 
neutrality, while on the average drifting in opposite directions according 
to Ohm's law under the action of electric fields. At higher frequencies the 
single-fluid model breaks down. The electrons and ions tend to move 
independently, and charge separations occur. These charge separations 
produce strong restoring forces. Consequently oscillations of an electro- 
static nature are set up. If a magnetic field is present, other effects occur. 
The electrons and ions tend to move in circular or helical orbits in the 
magnetic field with orbital frequencies given by 

co B = — (10.87) 

mc 

When the fields are strong enough or the densities low enough that the 
orbital frequencies are comparable to the collision frequency, the concept 
of a scalar conductivity breaks down and the current flow exhibits a 
marked directional dependence relative to the magnetic field (see Problem 
10.5). At still higher frequencies the greater inertia of the ions implies that 
they will be unable to follow the rapid fluctuation of the fields. Only the 
electrons partake in the motion. The ions merely provide a uniform back- 
ground of positive charge to give electrical neutrality on the average. The 
idea of a uniform background of charge, and indeed the concept of an 
electron fluid, is valid only when we are considering a scale of length which 
is at least large compared to interparticle spacings (/ > n ~ lyi ). In fact, there 
is another limit, the Debye screening length, which for plasmas at reasonable 
temperatures is greater than « ~^, and which forms the actual dividing 
line between small-scale individual-particle motion and collective fluid 
motion (see the following section). 

To avoid undue complications we consider only the high-frequency 
behavior of a plasma, ignoring the dynamical effects of the ions. We also 
ignore the effects of collisions. The electrons of charge e and mass m are 



336 



Classical Electrodynamics 



described by a density n(x, t) and an average velocity v(x, t). The equi- 
librium-charge density of ions and electrons is ^en^. The dynamical 
equations for the electron fluid are 



dt 



(m) = 



By 
Bt 



+ (vV)v 



m \ 



E + - x B 

c 



--^v P 



mn 



(10.88) 



where the effects of the thermal kinetic energy of the electrons are described 

by the electron pressure p (here assumed a scalar). The charge and 

current densities are : 

p =e(n-n )\ 

(10.89) 
J = ens 



Thus Maxwell's 


equation 


s can be written 




V- 


E = 4-nein — n ) 




V 


•B = 




V 


c dt 




V 


_ 1 dE Anen 
x B = 

c dt c 



(10.90) 



We now assume that the static situation is the electron fluid at rest with 
n — n and no fields present, and consider small departures from that 
state due to some initial disturbance. The linearized equations of motion 
are 

dt 



mn \on/o 



d\ e 
dt m 
V • E - A-nen = 



47ren 



U = 



(10.91) 



VxB-i^- 

c dt 

plus the two homogeneous Maxwell's equations. Here «(x, t) and v(x, t) 
represent departures from equilibrium. If an external magnetic field B 
is present a [(v/c) x B ] term must be kept in the force equation (see 
Problem 10.7), but the fluctuation field B is of first order in small quantities 
so that (v x B) is second order. The continuity equation is actually not 



[Sect. 10.9] Magnetohydrodynamics and Plasma Physics 337 

an independent equation, but may be derived by combining the last two 
equations in (10.91). 

Since the force equation in (10.91) is independent of magnetic field, we 
suspect that there exist solutions of a purely electrostatic nature, with 
B = 0. The continuity and force equations can be combined to yield a 
wave equation for the density fluctuations : 

?» + l*z£»*\ n _ 1 (?P\ V 2 n = (10 .92) 

dt 2 \ m / m \9n/o 

On the other hand, the time derivative of Ampere's law and the force 
equation can be combined to give an equation for the fields : 

^E /W«p\ E _ 1 (dp\ V(y . E) = cV x » (10 .93) 
dt 2 \ m 1 m \dnJo ot 

The structures of the left-hand sides of these two equations are essentially 
identical. Consequently no inconsistency arises if we put dB/dt = 0. 
Having excluded static fields already, we conclude that B = is a possi- 
bility. If dB/dt = 0, then Faraday's law implies V x E = 0. Hence E is 
a longitudinal field derivable from a scalar potential. It is immediately 
evident that each component of E satisfies the same equation (10.92) as the 
density fluctuations. If the pressure term in (10.92) is neglected, we find 
that the density, velocity, and electric field all oscillate with the plasma 
frequency (o p : 

2 _ 47rn o g (10.94) 



«V = 

m 



If the pressure term is included, we obtain a dispersion relation for the 
frequency : 

co 2 = co/ + - &) k 2 (10.95) 

m \dn/o 

The determination of the coefficient of k 2 takes some care. The adiabatic 
law p = p (nln o y can be assumed, but the customary acoustical value 
y = f for a gas of particles with 3 external, but no internal, degrees of 
freedom is not valid. The reason is that the frequency of the present 
density oscillations is much higher than the collision frequency, contrary 
to the acoustical limit. Consequently the one-dimensional nature of the 
density oscillations is maintained. A value of y appropriate to 1 trans- 
lational degree of freedom must be used. Since y = (m + 2)//w, where m 
is the number of degrees of freedom, we have in this case y = 3. Then 



i (?p) _ 

m \dn'o 



3 -P»_ (10.96) 

mn 



338 



Classical Electrodynamics 



If we use/? = n KT and define the rms velocity component in one direction 
(parallel to the electric field), 



m(u 2 ) = KT 
then the dispersion equation can be written 

a, 2 = co p 2 + 3(u 2 )k 2 



(10.97) 



(10.98) 



This relation is an approximate one, valid for long wavelengths, and is 
actually just the first two terms in an expansion involving higher and 
higher moments of the velocity distribution of the electrons (see Problem 
10.6). In form (10.98) the dispersion equation has a validity beyond the 
ideal gas law which was used in the derivation. For example, it applies to 
plasma oscillations in a degenerate Fermi gas of electrons in which all cells 
in velocity space are filled inside a sphere of radius equal to the Fermi 
velocity V F . Then the average value of the square of a component of 
velocity is 

< M 2 > = lV r * (10.99) 

Quantum effects appear explicitly in the dispersion equation only in higher- 
order terms in the expansion in powers of k 2 . 

The oscillations described above are longitudinal electrostatic oscilla- 
tions in which the oscillating magnetic field vanishes identically. This 
means that they cannot give rise to radiation in an unbounded plasma. 
There are, however, modes of oscillation in a plasma which are transverse 
electromagnetic waves. To see the various possibilities of plasma oscil- 
lations we assume that all variables vary as exp (ik • x — icot) and look for 
a defining relationship between co and k, as we did for the magnetohydro- 
dynamic waves in Section 10.8. With this assumption the linearized 
equations (10.91) and the two homogeneous Maxwell's equations can be 
written : 



k-v 

n = n 



ieE , 3<w 2 > n . 
mco co n„ 



'0 



k • E = — iArren 
k-B = 
k x B = 



— E — i ^v 



co 



k x E = -B 

c 



(10.100) 



[Sect. 10.10] Magnetohydrodynamics and Plasma Physics 339 

Maxwell's equations can be solved for v in terms of k and E : 

v = (-^-) —, - [(co 2 - c 2 /c 2 )E + c 2 (k • E)k] (10.101) 

\mtol co p 

Then the force equation and the divergence of E can be used to eliminate v 
in order to obtain an equation for E alone : 

(co 2 - co/ - c 2 fc 2 )E + (c 2 - 3<M 2 »(k • E)k = (10.102) 
If we write E in terms of components parallel and perpendicular to k: 

E = E|j -f- Ej_ 

where E„ = H^r) 1 

then (10.102) can be written as two equations: 

(co 2 - co/ - 3<« 2 )A: 2 )E|, = 



(10.103) 



(10.104) 
(co 2 - ft) p 2 - c 2 & 2 )Ej. = ' 

The first of these results shows that the longitudinal waves satisfy the 
dispersion relation (10.98) already discussed, while the second shows that 
there are two transverse waves (two states of polarization) which have the 
dispersion relation : 

co 2 = co/ + c 2 A: 2 (10.105) 

Equation (10.105) is just the dispersion equation for the transverse 
electromagnetic waves described in Section 7.9 from another point of view. 
In the absence of external fields the electrostatic oscillations and the trans- 
verse electromagnetic oscillations are not coupled together. But in the 
presence of an external magnetic induction, for example, the force equa- 
tion has an added term involving the magnetic field and the oscillations 
are coupled (see Problem 10.7). 



10.10 Short-Wavelength Limit for Plasma Oscillations and the 
Debye Screening Distance 

In the discussion of plasma oscillations so far no mention has been made 
of the range of wave numbers over which the description in terms of 
collective oscillations applies. Certainly n£ is one upper bound on the 
wave-number scale. A clue to a more relevant upper bound can be 
obtained by examining the dispersion relation (10.98) for the longitudinal 
oscillations. For long wavelengths the frequency of oscillation is very 



340 



Classical Electrodynamics 



closely co = co p . It is only for wave numbers comparable to the Debye 



wave number k 



d-> 



k D * = 



<u 2 > 



(10.106) 



that appreciable departures of the frequency from co p occur. 

For wave numbers k < k D , the phase and group velocities of the 
longitudinal plasma oscillations are : 



k 

3<« 2 > 



(10.107) 



From the definition of k D we see that for such wave numbers the phase 
velocity is much larger than, and the group velocity much smaller than, 
the rms thermal velocity (u 2 ) 1A . As the wave number increases towards 
k D , the phase velocity decreases from large values down towards (u 2 ) 1/4 . 
Consequently for wave numbers of the order of k D the wave travels with 
a small enough velocity that there are appreciable numbers of electrons 
traveling somewhat faster than, or slower than, or at about the same speed 
as, the wave. The phase velocity lies in the tail of the thermal distribution. 
The circumstance that the wave's velocity is comparable with the electronic 
thermal velocities is the source of an energy-transfer mechanism which 
causes the destruction of the oscillation. The mechanism is the trapping 
of particles by the moving wave with a resultant transfer of energy out of 
the wave motion into the particles. The consequent damping of the wave 
is called Landau damping. 

A detailed calculation of Landau damping is out of place here. But we 
can describe qualitatively the physical mechanism. Fig. 10.13 shows a 
distribution of electron velocities with a certain rms spread and a 
Maxwellian tail out to higher velocities. For small k the phase velocity 



no(v) 




Fig. 10.13 Thermal velocity 
distribution of electrons. 



[Sect. 10.10] Magnetohydrodynamics and Plasma Physics 341 

lies far out on the tail and negligible damping occurs. But as k -> k D the 
phase velocity lies within the tail, as shown in Fig. 10.13, with a significant 
number of electrons having thermal speeds comparable to v p . There is 
then a velocity band Ay around v = v p where electrons are moving 
sufficiently slowly relative to the wave that they can be trapped in the 
potential troughs and carried along at velocity v p by the wave. If there are 
more particles in At; moving initially slower than v p than there are 
particles moving faster (as shown in the figure), the trapping process will 
cause a net increase in the energy of the particles at the expense of the 
wave. This is the mechanism of Landau damping. Detailed calculations 
show that the damping can be expressed in terms of an imaginary part of 
the frequency given by 

Im co ~ -co p J"- (^>Je-«» 2 <^ (10.108) 



provided k<k D . To obtain (10.108) a Maxwellian distribution of 
velocities was assumed. For k >k D the damping constant is larger than 
given by (10.108) and rapidly becomes much larger than the real part of 
the frequency, as given by (10.98). 

The Landau formula (10.108) shows that for k<k D the longitudinal 
plasma oscillations are virtually undamped. But the damping becomes 
important as soon as k~k D (even for k = 0.5^, Im co ~ — 0Jco p ). 
For wave numbers larger than the Debye wave number the damping is so 
great that it is meaningless to speak of organized oscillations. 

Another, rather different consideration leads to the same limiting Debye 
wave number as the boundary of collective oscillatory effects. We know 
that an electronic plasma is a collection of electrons with a uniform back- 
ground of positive charge. On a very small scale of length we must 
describe the behavior in terms of a succession of very many two-body 
Coulomb collisions. But on a larger scale the electrons tend to cooperate. 
If a local surplus of positive charge appears anywhere, the electrons rush 
over to neutralize it. This collective response to charge fluctuations is 
what gives rise to large-scale plasma oscillations. But in addition to, or, 
better, because of, the collective oscillations the cooperative response of 
the electrons also tends to reduce the long-range nature of the Coulomb 
interaction between particles. An individual electron is, after all, a local 
fluctuation in the charge density. The surrounding electrons are repelled 
in such a way that they tend to screen out the Coulomb field of the chosen 
electron, turning it into a short-range interaction. That something like 
this must occur is obvious when one realizes that the only source of 
electrostatic interaction is the Coulomb force between the particles. If 
some of it is effectively taken away to cause long-wavelength collective 



342 Classical Electrodynamics 

plasma oscillations, the residue must be a sum of short-range interactions 
between particles. 

A nonrigorous derivation of the screening effect described above was 
first given by Debye and Hiickel in their theory of electrolytes. The basic 
argument is as follows. Suppose that we have a plasma with a distribution 
of electrons in thermal equilibrium in an electrostatic potential O. Then 
they are distributed according to the Boltzmann factor e ~ H/KT where His 
the electronic Hamiltonian. The spatial density of electrons is therefore 

n(x) = n e-W KT > (10.109) 

Now we imagine a test charge Ze placed at the origin in this distribution 
of electrons with its uniform background of positive ions (charge density 
— en ). The resulting potential <I> will be determined by Poisson's equation 

V 2 = -AirZe <5(x) - 4>nen Q [e-^ IKT ) - 1] (10.110) 

If (e<S>/KT) is assumed small, the equation can be linearized: 

V a O - k 2 D ® ~ -AirZe d(x) (10. 1 1 1) 

where 

, 2 _ 47rn e 2 /inii<y> 

k d — KT (10.112) 

is an alternative way of writing (10.106). Equation (10.111) has the 
spherically symmetric solution : 



-k D r 



<&(r) = Ze (10.113) 

r 

showing that the electrons move in such a way as to screen out the Coulomb 
field of a test charge in a distance of the order of k D ~ x . The balance between 
thermal kinetic energy and electrostatic energy determines the magnitude 
of the screening radius. Numerically 



\n / 



cm (10.114) 



where T is in degrees Kelvin, and n is the number of electrons per cubic 
centimeter. For a typical hot plasma with T = 10 6 °^and n = 10 15 cm -3 , 
we find kjf 1 ^ 2.2 x 10~ 4 cm. 

For the degenerate electron gas at low temperatures the Debye wave 
number k D is replaced by a Fermi wave number k F : 



k F ~^f- (10.115) 



where V F is the velocity at the surface of the Fermi sphere. This magni- 
tude of screening radius can be deduced from a Fermi-Thomas generaliza- 
tion of the Debye-Huckel approach. It fits in naturally with the dispersion 
relation (10.98) and the mean square velocity (10.99). 



[Probs. 10] Magnetohydrodynamics and Plasma Physics 343 

The Debye-Huckel screening distance provides a natural dividing line 
between the small-scale collisions of pairs of particles and the large-scale 
collective effects such as plasma oscillations. It is a happy and not 
fortuitous happening that plasma oscillations of shorter wavelengths can 
independently be shown not to exist because of severe damping. 



REFERENCES AND SUGGESTED READING 

The subject of magnetohydrodynamics and plasma physics has a rapidly growing 
literature. Many of the available books are collections of papers presented at confer- 
ences and symposia. Although these are useful to someone who has some knowledge of 
the field, they are not suited for beginning study. Two works on magnetohydrodynamics 
which are coherent presentations of the subject are 

Alfv6n, 

Cowling. 
A short discussion of magnetohydrodynamics appears in 

Landau and Lifshitz, Electrodynamics of Continuous Media, Chapter VIII. 
Corresponding books devoted mainly to the physics of plasmas are 

Chandrasekhar, 

Linhart, 

Simon, 

Spitzer. 
The subject of controlled thermonuclear reactions, with much material on the funda- 
mental physics of plasmas, is treated thoroughly by 

Glasstone and Lovberg, 

Rose and Clark. 



PROBLEMS 

10.1 An infinitely long, solid, right circular, metallic cylinder has radius (R/2) 
and conductivity a. It is tightly surrounded by, but insulated from, a hollow 
cylinder of the same material of inner radius (R/2) and outer radius R. 
Equal and opposite total currents, distributed uniformly over the cross- 
sectional areas, flow in the inner cylinder and in the hollow outer one. 
At t = the applied voltages are short-circuited. 

(a) Find the distribution of magnetic induction inside the cylinders 
before t = 0. 

(b) Find the distribution as a function of time after t = 0, neglecting 
the displacement current. 

(c) What is the behavior of the magnetic induction as a function of time 
for long times? Define what you mean by long times. 

10.2 A comparatively stable self-pinched column of plasma can be produced by 
trapping an axial magnetic induction inside the plasma before the pinch 
begins. Suppose that the plasma column initially fills a conducting tube of 
radius R and that a uniform axial magnetic induction B z0 is present in the 



344 Classical Electrodynamics 

tube. Then a voltage is applied along the tube so that axial currents flow 
and an azimuthal magnetic induction is built up. 

(a) Show that, if quasi-equilibrium conditions apply, the pressure-balance 
relation can be written : 



pir) + £l + £*! 



1 ["Eldr 



+ — t±- dr = 



(b) If the plasma has a sharp boundary and such a large conductivity 
that currents flow only in a thin layer on the surface, show that for a 
quasi-static situation the radius R(t) of the plasma column is given by the 
equation 



where 



W£)-f.I>* 



BmRft 

'0 — 



cE 

and E f(t) is the applied electric field. 

(c) If the initial axial field is 100 gauss, and the applied electric field has 
an initial value of 1 volt/cm and falls almost linearly to zero in 1 millisecond, 
determine the final radius if the initial radius is 50 cm. These conditions are 
of the same order of magnitude as those appropriate for the British toroidal 
apparatus (Zeta), but external inductive effects limit the pinching effect to 
less than the value found here. See E. P. Butt et al., Proceedings of the 
Second International Conference on Peaceful Uses of Atomic Energy, Vol. 32, 
p. 42 (1958). 

10.3 Magnetohydrodynamic waves can occur in a compressible, nonviscous, 
perfectly conducting fluid in a uniform static magnetic induction B . If the 
propagation direction is not parallel or perpendicular to B , the waves are 
not separated into purely longitudinal (magnetosonic) or transverse (Alfven) 
waves. Let the angle between the propagation direction k and the field B 
bed. 

(a) Show that there are three different waves with phase velocities given by 

«i 2 = (Vx cos 0) 2 

"1,3 = K* 2 + *>a 2 ) ± U(s 2 + va 2 ) 2 - 4sW cos 2 6}V2 

where s is the sound velocity in the fluid, and va. = (B 2 /47rp o yA is the 
Alfven velocity. 

(b) Find the velocity eigenvectors for the three different waves, and 
prove that the first (Alfven) wave is always transverse, while the other two 
are neither longitudinal nor transverse. 

(c) Evaluate the phase velocities and eigenvectors of the mixed waves in 
the approximation that vj, :> s. Show that for one wave the only appreciable 
component of velocity is parallel to the magnetic field, while for the other 
the only component is perpendicular to the field and in the plane containing 
k and B . 

10.4 An incompressible, nonviscous, perfectly conducting fluid with constant 
density p is acted upon by a gravitational potential y> and a uniform, static, 
magnetic induction B . 



[Probs. 10] Magnetohydrodynamics and Plasma Physics 345 

(a) Show that magnetohydrodynamic waves of arbitrary amplitude and 
form BjCx, 0, v(x, t) can exist, described by the equations 

(B -V)B 1 = ±V47r Po — x 

B x = ±V4^v 

, (B + B x ) 2 t + 

P + PaW H ^ — — = constant 

(Z>) Suppose that at r = a certain disturbance B^x, 0) exists in the 
fluid such that it satisfies the above equations with the upper sign. What 
is the behavior of the disturbance at later times ? 
10.5 The force equation for an electronic plasma, including a phenomenological 
collision term, but neglecting the hydrostatic pressure (zero temperature 
approximation) is 



^ + (v . V )v =-U+- xb) -v 
dt m\ c J 



where v is the collision frequency. 

(a) Show that in the presence of static, uniform, external, electric, and 
magnetic fields, the linearized steady-state expression for Ohm's law 
becomes 

i 



where the conductivity tensor is 



1 ^ 



,2\| V 



4ttvI 1 + - B ' 
\ v 



|i +^4 



and co v (to B ) is the electronic plasma (precession) frequency. The direction 
of B is chosen as the z axis. 

(b) Suppose that at t = an external electric field E is suddenly applied 
in the x direction, there being a magnetic induction B in the z direction. 
The current is zero at t = 0. Find expressions for the components of the 
current at all times, including the transient behavior. 
10.6 The effects of finite temperature on a plasma can be described approxi- 
mately by means of the correlationless Boltzmann (Vlasov) equation. Let 
f(x, v, t) be the distribution function for electrons of charge e and mass m 
in a one-component plasma. The Vlasov equation is 

f=f + v.v,/ + a.V„/ = 

where V^ and V v are gradients with respect to coordinate and velocity, and 
a is the acceleration of a particle. For electrostatic oscillations of the 



346 Classical Electrodynamics 

plasma a = eE/m, where E is the macroscopic electric field satisfying 

V • E = 4ne I /(x, v, /) (Pv - n 

If / (v) is the normalized equilibrium distribution of electrons 

/ (v) d 3 v = n 



"o /o( 



(a) show that the dispersion relation for small-amplitude longitudinal 
plasma oscillations is 



co/ Jk-v-o) 



(b) assuming that the phase velocity of the wave is large compared to 
thermal velocities, show that the dispersion relation gives 

0)2 i ^^< k * v > , , <(k-v) 2 > , 

5 ^ 1 + l + 5 5 — + • • • 

co/ co or 

where < > means averaged over the equilibrium distribution / (v). Relate 
this result to that obtained in the text with the electronic fluid model. 

(c) What is the meaning of the singularity in the dispersion relation when 
k-v = co? 

10.7 Consider the problem of waves in an electronic plasma when an external 
magnetic field B is present. Use the fluid model, neglecting the pressure 
term as well as collisions. 

(a) Write down the linearized equations of motion and Maxwell's 
equations, assuming all variables vary as exp (ik '• x — ia>t). 

(b) Show that the dispersion relation for the frequencies of the different 
modes in terms of the wave number can be written 

co 2 (co 2 - VX^ 2 - V - * 2c2 ) 2 

= oo B 2 (a>* - Ar 2 c 2 )[o> 2 (co 2 - co/ - A: 2 c 2 ) + co/c^k • b) 2 ] 

where b is a unit vector in the direction of B ; co v and a> B are the plasma 
and precession frequencies, respectively. 

(c) Assuming co B <; a> p , solve approximately for the various roots for 
the cases (i) k parallel to b, (ii) k perpendicular to b. Sketch your results 
for co 2 versus k 2 in the two cases. 



11 



Special Theory of Relativity 



The special theory of relativity has been treated extensively in many 
books. Its history is interwoven with the history of electromagnetism. In 
fact, one can say that the development of Maxwell's equations with the 
unification of electricity and magnetism and optics forced special relativity 
on us. Lorentz laid the groundwork in his studies of electrodynamics, 
while Einstein contributed crucial concepts and placed the theory on a 
consistent and general basis. Even though special relativity had its origin 
in electromagnetism and optics, it is now believed to apply to all 
types of interactions except, of course, large-scale gravitational phenomena. 
In modern physics the theory serves as a touchstone for possible forms for 
the interactions between elementary particles. Only theories consistent 
with special relativity need to be considered. This often severely limits the 
possibilities. Since the experimental basis and the development of the 
theory are described in detail in many places, we will content ourselves 
with a summary of the key points. 



11.1 Historical Background and Key Experiments 

In the forty years before 1900 electromagnetism and optics were cor- 
related and explained in triumphal fashion by the wave theory based on 
Maxwell's equations. Since previous experience with wave motion had 
always involved a medium for the propagation of waves, it was natural for 
physicists to assume that light needed a medium through which to propa- 
gate. In view of the known facts about light it was necessary to assume 
that this medium, called the ether, permeated all space, was of negligible 
density, and had negligible interaction with matter. It existed solely as a 
vehicle for the propagation of electromagnetic waves. The hypothesis of 

347 



348 Classical Electrodynamics 

an ether set electromagnetic phenomena apart from the rest of physics, 
For a long time it had been known that the laws of mechanics were the 
same in different coordinate systems moving uniformly relative to one 
another — the laws of mechanics are invariant under Galilean transfor- 
mations. The existence of an ether implied that the laws of electro- 
magnetism were not invariant under Galilean coordinate transformations. 
There was a preferred coordinate system in which the ether was at rest. 
There the velocity of light in vacuum was equal to c. In other coordinate 
frames the velocity of light was presumably not c. 

To avoid setting electromagnetism apart from the rest of physics by a 
failure of Galilean relativity there are several avenues open. Some of these 
are: 

1. Assume that the velocity of light is equal to c with respect to a 
coordinate system in which the source is at rest. 

2. Assume that the preferred reference frame for light is the coordinate 
system in which the medium through which the light is propagating is at 
rest. 

3. Assume that, although the ether has a very small interaction with 
matter, the interaction is enough that it can be carried along with astro- 
nomical bodies such as the earth. 

A large number of experiments led to the abandonment of all these 
hypotheses and the birth of the special theory of relativity. Three basic 
experiments are : 

(1) Observation of the aberration of star positions during the year, 

(2) Fizeau's experiment on the velocity of light in moving fluids (1859), 

(3) Michel son-Morley experiment to detect motion through the ether 
(1887). 

The aberration of star light (the small shift in apparent position of 
distant stars during the year) is an ancient phenomenon which finds a 
simple explanation in the motion of our earth in its orbit around the sun 
at a velocity of the order of 3 x 10 6 cm/sec. Suppose that the star light 
is incident normal to the earth's surface while the velocity of the earth in 
orbit is parallel to the surface. Figure 11.1 shows that a telescope must be 
inclined at an angle a, where 

v 



a 



~ - ~ 1(T 4 radian (11.1) 



c 



in order that the light pass down it to the observer as the telescope moves 
along. Six months later the velocity vector v will be in the opposite 
direction. The star will then appear at an angle a on the other side of the 
vertical. The apparent positions of stars trace out elliptical paths on the 
celestial sphere during the year with angular spreads of the order of (11.1). 



[Sect. 11.1] 



Special Theory of Relativity 



349 




Fig. 11.1 Aberration of star positions. 



This simple explanation of aberration contradicts the hypothesis that the 
velocity of light is determined by the transmitting medium (our atmosphere 
in this case) or that the ether is dragged along by the earth. In neither case 
would aberration occur. 

Fizeau's experiment involved measuring, by means of an interferometer, 
the velocity of light in liquids flowing in a pipe, both in the direction of and 
opposed to the propagation of the light. If the index of refraction of the 
liquid is n, then depending on which of the various hypotheses one 
chooses, he expects the velocity to be 



c 

u = -, 
n 



n 



(11.2) 



where v is the velocity of flow. The actual result found by Fizeau was, 
within experimental error, 



B -£ ± ,(i_l) 

n \ n I 



(11.3) 



We note that this result can be made consistent with the ether being dragged 
along by the earth only by assuming that smaller bodies are partially 
successful in carrying the ether with them. Even then the assumption is 
rather artificial in that their effectiveness at carrying the ether involves 
their indices of refraction. * 

The Michelson-Morley experiment was designed to detect a motion of 
the earth relative to a preferred reference frame (the ether at rest) in which 
the velocity of light is c. The basic apparatus is shown schematically in 
Fig. 1 1.2. A laboratory light source S is focused on a thinly silvered glass 
plate P which divides the light into two beams at right angles to each other, 
one of which goes to mirror M x and is reflected back through the plate 



* Actually formula (11.3) is a theoretical one proposed in 1818 by Fresnel on the basis 
of a model in which the density of the elastic ether in matter is proportional to n 2 . 



350 



Classical Electrodynamics 




B 2 By 



Fig. 11.2 Michelson-Morley experiment. 



to B x and the other of which goes to mirror M 2 , back to the plate, and is 
reflected to B 2 . Conditions are such that the two beams travel almost the 
same path length. Small differences in path length or in the times taken to 
traverse the paths are detected by observing shifts in interference fringes 
produced by the two beams. The whole apparatus was attached to a stone 
slab floating in mercury so that it could be rotated. Suppose that velocity 
v of the earth through the ether is parallel to the light path from P to M 2 . 
Then the velocity of light relative to the apparatus on the path from P to 
M 2 and return is c ± v. If the path distance from P to M 2 is d 2 , the time 
taken by the light to go from P to M 2 and return is 



-4— + -T-)-* 

\c — V C + VI c 



1 



(-3 



(11.4) 



For the path from P to M x and return it is convenient to view things from 
the preferred coordinate frame. Then it is evident that the path length 
traversed is greater than d lf the distance from P to M x , because the mirror 
is moving with velocity v through the ether. 

Figure 11.3 shows the geometrical relations. Evidently sin a = v/c, so 
that the effective path length is 



2d x sec a = 2d x 



1 



i-4 

c 



and the time taken is 



2d x 1 



(11.5) 



[Sect. 11.1] Special Theory of Relativity 

The difference between the two transit times is 



At = t 2 — t x = - 





d 2 


c 


k 






v 2 


1 




f7 2 


1 




J 


— 






c 2 ' 


V 




c 2 -l 



351 



(11.6) 



If we assume that v < c, we can expand the denominators, obtaining 



At 



(11.7) 



If the apparatus is now rotated through 90°, the transit times become 



H — 

c 



f ,- 2d i 

h — 

c 



1 



J*-$ 



(11.8) 



1 



i-4 



and the difference, to lowest order, is 



At' - ?[(<*, - d x ) + ^(| - ^)] (11.7') 



Since At and At' are not the same, we expect a shift in the interference 
fringes upon rotation of the apparatus, the shift being proportional to 

t 

„2 



At' — At = 



1 17" 

-(di + dj- 
c c l 



(11.9) 



Since the orbital velocity of the earth is about 3 X 10 6 cm/sec, we expect 
v 2 /c z ~ 10~ 8 , at least at some time of the year. With (d x + d^ ~ 3 x 10 2 
cm, the time difference (11.9) is 10 -16 sec. This means that the relevant 
length (to be compared to a wavelength of light) is c \At' — Ar| . — ' 3 x 
10 -6 cm = 300 A. Since visible light has wavelengths of the order of 



CZZ.3 




Fig. 11.3 



352 Classical Electrodynamics 

3000 A, the expected effect is a fringe shift of about one-tenth of a fringe. 
The accuracy of the Michelson-Morley experiment was such that a 
relative velocity of 10 6 cm/sec would have been seen (i.e., one-third of the 
above estimate). No fringe shift was found. Since the original work of 
Michelson the experiment has been repeated many times with modifi- 
cations such as very unequal path lengths. No evidence for relative motion 
through the ether has been found. A summary of all the available evidence 
has been given by Shankland et ah, Revs. Modern Phys., 27, 167 (1955). 

The negative result of the Michelson-Morley experiment can be ex- 
plained on the ether-drag hypothesis. But that hypothesis is inconsistent 
with the aberration of star light. Only the so-called emission theories, 
where the velocity of light is constant relative to the source, are consistent 
with all three of the experiments cited. And we will see in the next section 
that other experiments exclude such theories. On the positive side the 
Michelson-Morley experiment can be thought of as restoring electro- 
magnetism to the rest of physics in the matter of relativity. No observable 
effects were found which depended on the motion of the apparatus 
relative to some conjectured absolute reference frame. It should be 
mentioned, however, that FitzGerald and Lorentz (1892) explained the 
null result while still retaining the ether concept by postulating that all 
material objects are contracted in their direction of motion as they move 
through the ether. The rule of contraction is 



UP)= L 0a/ 1_ ~2 (1L10) 

It is clear from (11.4) or (11.7) that this hypothesis leads to a zero result 
for (At' — At) in place of (11.9). The FitzGerald-Lorentz contraction 
hypothesis was perhaps the last gasp of the ether advocates, and it contains 
the germ of the special theory of relativity. The contraction phenomenon 
is present in special relativity, but in a more general way applying to all 
systems in relative motion with one another. Going along with it is the 
phenomenon of time dilatation (not postulated by FitzGerald or Lorentz), 
an experimentally well-founded effect. These are discussed in Section 1 1.3. 



11.2 The Postulates of Special Relativity and the Lorentz 
Transformation 

In 1904 Lorentz showed that Maxwell's equations in vacuum were 
invariant under a transformation of coordinates given by (11.19) below, 
and now called a Lorentz transformation, provided the field strengths were 



[Sect. 11.2] Special Theory of Relativity 353 

suitably transformed. By supposing that all matter was essentially electro- 
magnetic in origin and so transformed in the same way as Maxwell's 
equations, Lorentz was able to deduce the contraction law (11.10). Then 
Poincare showed that the transformation of charge and current densities 
could be made in such a way that all the equations of electrodynamics are 
invariant in form under Lorentz transformations. In 1905, almost at the 
same time as Poincare and without knowledge of Lorentz's paper, Einstein 
formulated special relativity in a general and complete way, obtaining the 
results of Lorentz and Poincare, but showing that the ideas were of much 
wider applicability. Instead of basing his discussion on electrodynamics, 
Einstein showed that just two postulates were necessary, one of them 
involving only a very general property of light. 
The two postulates of Einstein were : 

1 . POSTULAtE OF RELATIVITY 

The laws of nature and the results of all experiments performed 
in a given frame of reference are independent of the translational 
motion of the system as a whole. Thus there exists a triply infinite 
set of equivalent reference frames moving with constant velocities 
in rectilinear paths relative to one another in which all physical 
phenomena occur in an identical manner. 

For brevity these equivalent coordinate systems are called Galilean 
reference frames. The postulate of relativity is consistent with all our 
experience with mechanics where only relative motion between bodies is 
relevant. It is also consistent with the Michelson-Morley experiment and 
makes meaningless the question of detecting motion relative to the ether. 

2. POSTULATE OF THE CONSTANCY OF THE VELOCITY OF LIGHT 

The velocity of light is independent of the motion of the source. 

This hypothesis, untested when Einstein proposed it, is necessary and 
decisive in obtaining the Lorentz transformation and all its consequences 
(see below). Because our classical concept of time as a variable independent 
of the spatial coordinates is destroyed by this postulate, its acceptance was 
resisted vehemently for a number of years. Many ingenious attempts were 
made to invent theories which would explain all the observed facts without 
this assumption. One notable one was Ritz's version of electrodynamics, 
which kept the two homogeneous Maxwell's equations intact but modified 
the equations involving the sources in such a way that the velocity of light 
was equal to c only when measured relative to the source. Experiments 
have proved all such theories wrong and have established the constancy 
of the velocity of light independent of the motion of the source. One such 
experiment is the Michelson-Morley interferometer experiment performed 



354 



Classical Electrodynamics 



o\ ) > 



Fig. 11.4 

with starlight, rather than a terrestrial light source. No effect was observed 
which could be attributed to a change in the velocity of light due to the 
relative motion of the star and the earth. Another experiment on the light 
from rotating binary stars showed that the velocity of light depends 
negligibly (if c' = c + kv, then k < 0.002) on the motion of the stars 
toward or away from us. 

The constancy of the velocity of light, independent of the motion of the 
source, allows us to deduce the connection between space-time coordinates 
in different Galilean reference frames. To see how this is possible we 
consider two coordinate systems K and K'. System K' has its axes parallel 
to those of K, but it is moving with a velocity v in the positive z direction 
relative to the system K, as shown in Fig. 11.4. Points in space and time 
in the two systems are specified by (x, y, z, t) and (x', y', z', t), respectively. 
For convenience we suppose that a common origin of time t — t ' = is 
chosen at the instant when the two sets of coordinate axes exactly overlap. 
Now we imagine an observer in each reference frame equipped with the 
necessary apparatus (e.g., a network of correlated clocks and photocells 
at known distances from the origin) to detect the arrival time of a light 
signal from the origin at various points in space. If there is a light source 
at rest in the system K (and so moving with velocity v in the negative z 
direction in system K') which is flashed on and off rapidly at t = t' = 0, 
then Einstein's second postulate implies that each observer will see his 
photocell network respond to a spherical shell of radiation moving out- 
ward from his origin of coordinates with velocity c. Consequently the 
arrival time / of the pulse at a detector located at (x, y, z) in system K will 
satisfy the equation : 



x 2 + y 2 + z 2 - c 2 t 2 = 
Similarly, in system K' the wave front is described by 

x' 2 + y' 2 + z' 2 - cH' 2 = 



(11.11) 



(11.12) 



[Sect.11.2] 



Special Theory of Relativity 



355 



Relations (11.11) and (11.12) seem to violate the first postulate of relativity. 
If two observers in different coordinate systems both see spherical pulses 
centered on a fixed origin in each system, the spheres must be different! 
This apparent contradiction is resolved when we allow the possibility that 
events which are simultaneous in one coordinate system are not necessarily 
simultaneous in another coordinate system moving relative to the first. We 
can now anticipate that time is no longer an absolute quantity independent 
of spatial variables and of relative motion. 

To obtain a connection between the coordinates (x\ y', z', t') of system 
K' and (x, y, z, i) of system Kit is only necessary to assume that the trans- 
formation is linear. This seems very plausible and is equivalent to the 
assumption that space-time is homogeneous and isotropic. If the trans- 
formation is linear, the only possible connection between the quadratic 
forms (11.11) and (11.12) is 

z' 2 + y' 2 + z' 2 - c 2 t' 2 = X\x 2 + y* + z 2- c 2 t 2 ) (11.13) 

where X = X{v) with X(0) = 1. The presence of X allows for the possibility 
of an overall scale change in going from K to K'. But shells of radiation 
are spheres in both systems. From the hypothesis that K' is moving 
parallel to the z axis of K, it is evident that the transformation of x', y' 
must be 

x' = Xx, y' = Xy (11.14) 

independent of the time, because motion parallel to the z axis in K' must 
remain so in K. Then the most general linear connection between z', t' 
and z, t is 

z' = X{a x z + a 2 t), t' = X(b x t + b#) (11.15) 

A factor X has been extracted for convenience. The coefficients a x , a 2 , b lt b 2 
are functions of v with the following limiting values as v — ► : 





«i 




V 


im . 


a% 


- — • 





;-+0 


h 




l 




A. 




0, 



(11.16) 



The origin of K' moves with a velocity v in the system K. Consequently 
its position is specified by z = vt. This means that a 2 = —va x in (11.15). 
If equations (11.15) are now substituted into (11.13), three algebraic 
relations between a x , b x , and b 2 are obtained. These are easily solved to 



356 Classical Electrodynamics 

give the following values, with signs chosen to agree with (11.16): 

, 1 

a x = b x = 



l- V - 
c 2 



(11.17) 



C 2 



There remains the problem of the determination of X(v). If a third 
reference frame K" is considered to be moving with a velocity — v parallel 
to the z axis relative to K\ the coordinates (x", y", z", t") can be obtained 
in terms of (x r , y', z', t') from the above results merely by the change 
v->—v. But the system K" is just the original system K, so that x" = x, 
y" = y, z" = z, t" = t. This leads to the requirement that 



X(v) X{-v) = 1 



(11.18) 



But X{v) must be independent of the sign of v, since it represents a scale 
change in the transverse direction. Consequently X(v) = 1. Then we can 
write down the Lorentz transformation, connecting coordinates in K' to 
those in K: 



t' = 



z — vt 



„2 



C 2 






x = x, y = y 



(11.19) 



Transformation (11.19) represents a special case in which the relative 
motion of K and K' is parallel to the z axis. It is a straight-forward matter 
to write down the result for an arbitrary velocity v of translation of K' 
relative to K, as shown in Fig. 11.5. Equation (11.19) clearly applies to 
parallel and perpendicular components of the coordinates relative to v: 



x„ = 



1 



(x„ - vf), x/ = x ± 



c 2 



j>4 cj 



(11.20) 



[Sect. 11.3] 



Special Theory of Relativity 



357 




Fig. 11.5 

With the definition that x„ = [(v • x)v]/y 2 and x ± = x — x„, equations 
(11.20) can be combined to yield the general Lorentz transformation:* 

1 Ax-v 1 



x' = x + 



- 1 






vt 



c 2 






M< - M 



l ~* 



(11.21) 



It should be noted that (11.21) represents a single Lorentz transformation 
to a reference frame K' moving with velocity v relative to the system K. 
Successive Lorentz transformations do not in general commute. It is easy 
to show that they commute only if the successive velocities are parallel. 
Consequently three successive transformations corresponding to the com- 
ponents of the velocity v in three mutually perpendicular directions yield 
different results, depending on the order in which the transformations are 
applied, and none agrees with (11.21). (See Problem 11.2.) 



11.3 FitzGerald-Lorentz Contraction and Time Dilatation 



As has already been mentioned, FitzGerald and Lorentz proposed the 
contraction rule (11.10) for the dimensions of an object parallel to its 

* The word "general" is not really applicable to transformation (1 1 .21). The connota- 
tion here is that the direction of the velocity v is arbitrary. But a more general trans- 
formation would allow the axes in K' to be rotated relative to those in, K. Even this 
Lorentz transformation is not the most general, since it is still homogeneous in the co- 
ordinates. The general inhomogeneous Lorentz transformation allows translation of the 
origin in space-time as well. See M0ller, Section 18. 



358 Classical Electrodynamics 



Fig. 11.6 Moving rod. FitzGerald-Lorentz 
contraction. 



motion at velocity v through the ether; Lorentz was able to give the rule 
an electrodynamic basis from the properties of Maxwell's equations under 
Lorentz transformations. We now show that the contraction of lengths 
in the direction of motion is a more general phenomenon which applies to 
all relative motion. Consider a rod of length L at rest parallel to the z' 
axis in the system K' of the previous section, as indicated schematically in 
Fig. 1 1.6. By definition L = z 2 ' — z/, where z/ and z 2 ' are the coordinates 
of the end points of the rod in K'. In the system A:the length L of the rod is, 
again by definition, L = z 2 — z x , where z 2 and z 1 are the instantaneous 
coordinates of the end points of the rod, observed at the same time t. 
From (11.19) the length in K' is 

L = V - z x > = Z )~ % \ = -== (11.22) 

which is just the FitzGerald-Lorentz result (11.10). Note that in the 
system K the length is defined at equal times t. The fact that this is not at 
equal times in the system K' is not relevant for the definition of length in 
the system K. This again illustrates that simultaneity is only a relative 
concept. 

Another consequence of the special theory of relativity is time dilatation. 
A clock moving relative to an observer is found to run more slowly than 
one at rest relative to him. The most fundamental "clocks" which we 
have available are the unstable elementary particles. Each type of particle 
decays at rest with a well-defined lifetime (mean life) which is unaltered by 
external fields, apart from nuclear or atomic force fields which cause 
transformations that are well understood.* These particles can serve 
therefore as "clocks" which can be examined at rest and in motion. 
Suppose that we consider a meson of lifetime t at rest in the system K' 

* For example, negative mu mesons can become bound in hydrogen-like orbits 
around nuclei with binding energies that are not negligible compared to the energy 
liberated in their decay. Since the rate of decay depends sensitively on the energy release 
(closely as the fifth power of AE), tightly bound negative mu mesons exhibit a consider- 
ably slower rate of decay than unbound ones. 



[Sect. 11.3] Special Theory of Relativity 359 

which is moving with uniform velocity v relative to the system K. We 

assume that the meson is created at the origin of K' at time t' = t = 0. As 

seen from the system K the position of the meson is given by z = vt. If it 

lives a time t in K', then at its instant of decay, we find 

v 
t -z 



c 2 L v 2 



i- v - 



''=-o = -l^= = 'Jl-- 9 (U- 23 ) 



c 

The time / is the meson's lifetime t as observed in the system K. Con- 
sequently 

T ° (11.24) 



c 2 

When viewed from K the moving meson lives longer than a meson at rest 
in K. The "clock" in motion is observed to run more slowly than an 
identical one at rest. 

Time dilatation has been observed in cosmic rays with high-energy mu 
mesons. These mesons are produced as secondary particles at a height of 
the order of 10 or 20 km, and a large fraction of them reach the earth's 
surface. Since the mean lifetime of a mu meson is t = 2.2 x 10~ 6 sec, 
it could travel no more than ct = 0.66 km on the average before decaying 
if no time dilatation occurred. Clearly dilatation factors of the order of 
10 or more are involved, consistent with the high energies (velocities 
approaching the velocity of light) of these particles. 

A laboratory experiment exhibiting time dilatation with pi mesons is 
not difficult to perform. Charged pi mesons have a mean lifetime t = 
2.56 x 10~ 8 sec. An experiment studying the numbers of charged pi 
mesons decaying in flight per unit length as a function of distance from the 
point of production was done at Columbia University. * The mesons had 
a velocity v ~ 0.75c. The numbers of mesons decaying per unit distance 
should follow an exponential law N(x) = N e~ x/X , where A is the mean 
free path in the laboratory and x is the distance from the source (corrected 
for finite solid angles, etc.). Figure 1 1.7 shows schematically the results of 
the experiment. The mean free path is X = 8.5 ± 0.6 meters. Since 
X = vt, the laboratory lifetime is r = 3.8 ± 0.3 x 10 _8 sec. Consequently 

r 2.56 

* Durbin, Loar, and Havens, Phys. Rev., 88, 179 (1952). This experiment was per- 
formed to measure the lifetime of the pi meson. The validity of time dilatation was 
assumed. But with independent knowledge of the lifetime, the argument can be inverted 
as we do here. 



360 



Classical Electrodynamics 




X obs = 8.5+0.6 
meters 



Fig. 11.7 Number of pi mesons decaying 
per unit distance as a function of distance 
from the point of production (schematic). 



This value is to be compared with 1.51 calculated from (11.24) with 
v = 0.75c. The laboratory experiment on time dilatation is not so 
dramatic as the cosmic rays but has the great virtue of being performed 
under controlled conditions in a comparatively small space. 



11.4 Addition of Velocities: Aberration and the Fizeau Experiment; 
Doppler Shift 



The Lorentz transformation (11.19) can be used to obtain the addition 
law for velocities. Suppose that there is a velocity vector u' in the system 
K' which makes polar angles 6', <f>' with the z' axis as shown in Fig. 11.8. 
The system K' is moving relative to the system K with a velocity v in the z 
direction. We want to know the components of velocity u as seen in the 
system K. From (11.19), or rather the inverse transformation, the dif- 
ferential expressions for dx, dy, dz, dt can be obtained: 



dx = dx', dy = dy' 
1 



dz = 



(dz' + v dt'), dt = 



c 2 



. 1 Lt' + -dz) 1(11 



25) 



This means that the components of velocity are 



u„ = 



1 - - "a 

c 2 



u z ' + v 



1 - 1 < 

c l 



1 + 



VIA, 



(11.26) 



[Sect. 11.4] 



Special Theory of Relativity 



361 



The magnitude of u and the angles 6, <f> of u in the system K are easily 
found. Since u x 'lu y ' = uju y , the azimuthal angles are equal, <f> = <f>'. 
Similarly 



tan0 = 



1-^ 



u sin 



c 2 u' cos 6' + v 



and 



+ v 2 + 2u'v cos 6' — 



u v sin 



1 + 



u ' v a>\ 

-cose) 



(11.27) 



The inverse results for (u', 0') in terms of (u, 6) can be obtained by inter- 
changing «<-> u', 6<-^ 6', and also changing the sign of v. 

Study of (11.26) or (11.27) shows that for velocities u' and v both small 
compared to c the addition law is just that of Galilean relativity, u = u' -f- 
v. But for either velocity approaching that of light, modifications appear. 
It is impossible to obtain a velocity greater than that of light by adding 
two velocities, even if each is very close to c. For the simple case of parallel 
velocities the addition law is 

+ v 



u = 



(11.28) 



1 + 



If u' = c, then (11.28) shows that u = c also. This is merely an explicit 
statement of Einstein's second postulate. 

The laws of addition of velocities are in accord with both the aberration 
of starlight and the Fizeau experiment. For the aberration, the velocity u' 




Fig. 11.8 Addition of velocities. 



362 Classical Electrodynamics 

is that of light in the system K', namely, u' = c, and v is the orbital velocity 
of the earth. Then the angle 6 is related to d' by 



«» 9 -v i -4- H!lL i (11 - 29) 

c cos 6' + - 
c 

For starlight incident normally on the earth 6' = tt/2. Then 



*™ d = C -Ji- v -2 01-30) 

The angle 6 is the complement of the angle a in (11.1). Thus 

tan a = —ML= (11.31) 

-$ 

completely consistent with observation. (Since vjc ~ 10 -4 , the departure 
of the radical from unity is far beyond the realm of observation.) 

In the simplest version of the Fizeau experiment the liquid flows with 
velocity v parallel or antiparallel to the path of light. If the liquid has an 
index of refraction n we may assume that light propagates with a velocity 
u' = c/n relative to the liquid. From (1 1 .28) the velocity of light observed 
in the laboratory is 

C -±v 
u = - - ± vl 1 - -U (11.32) 

nc 

The latter expression is the expansion to lowest order of the exact result. 
This is in agreement with the Fresnel formula (1 1.3). Actually there is an 
added term in (1 1.32) if the index of refraction is a function of wavelength. 
It comes about because of the Doppler shift in wavelength in the moving 
liquid. The increase AA in wavelength in the moving medium is 

AA=±An- (11.33) 

c 

correct to lowest order in vjc for the parallel and antiparallel velocities, 
respectively. Consequently the appropriate velocity of light in the liquid 
is 

c c c dn A . 

~ — Ax 

n(A + AA) n(X) n 2 dk 



[Sect. 11.4] Special Theory of Relativity 363 

Then the corrected expression to replace (11.32) is 

,— e ±„(l -!-*£) dl-34) 

The correction due to dispersive effects has been observed. 

The relativistic Doppler shift formulas can be obtained from the fact 
that the phase of a light wave is an invariant quantity. Actually, the phase of 
any plane wave is invariant under a Lorentz transformation, the reason 
being that the phase can be associated with mere counting which is 
independent of coordinate frame. Consider a plane wave of frequency co 
and wave vector k in the reference frame K. An observer at the point P 
with coordinate x is equipped to record the number of wave crests which 
reach him in a certain time. If the wave crest passing the origin at t = is 
the first one which he records (when it reaches him), then at time t he will 
have counted 

— (k • x — cot) 

2tt 

wave crests. Now imagine another reference frame K' which moves 
relative to the frame K with a velocity v parallel to the z axis, and has its 
origin coincident with that of K at t = 0. An observer in K' at the point 
P' with coordinate x' is equipped similarly to the one in K. He begins 
counting when the wave crest passing the origin reaches him, and con- 
tinues counting until time t '. If the point P' is such that at the end of the 
counting period it coincides with the point P, then both observers must have 
counted the same number of wave crests. But the observer in K' has 
counted 

wave crests, where k' and co' are the wave vector and frequency of the plane 
wave in K '. Thus the phase of the wave is invariant. Consequently we 
have 

k'-x' - co't' = k-x- cot (11.35) 

Using the transformation formulas (11.19), we find 

v ' — v U .' _ u 



■-tt'^ "fl 



2 (« - vk z ) 



(11.36) 



364 Classical Electrodynamics 

For light waves, |k| = cojc and |k'| = co'/c. Hence these results can be 
expressed in the form : 

CO 



HH 



c 2 



tan 6' = 



1 v 2 ■ 
1 - — sin 

c 2 

a v 
COS 



(11.37) 



c 

where 6 and 6' are the angles of k and k' relative to the direction of v. This 
last equation is just the inverse of (11.29). 

It is sometimes useful to have the frequency co' expressed in terms of the 
angle 0' of the wave in the frame K'. From the inverse of the first equation 
in (11.37), it is evident that the desired expression is 

i ^ 
1 -co 

C (11.38) 



(l+-COS0'j 

The first equation in (11.37) is the customary Doppler shift, modified 
by the radical in the denominator. The presence of the square root shows 
that there is a transverse Doppler shift, even when 6 = tt\2. This relati- 
vistic transverse Doppler shift has been observed spectroscopically with 
atoms in motion (Ives-Stilwell experiment, 1938). It also has been observed 
using a precise resonance-absorption technique, with a nuclear gamma-ray 
source on the axis of a rapidly rotating cylinder and the absorber attached 
to the circumference of the cylinder. 



11.5 Thomas Precession 

In 1926, Uhlenbeck and Goudsmit introduced the idea of electron spin 
and showed that, if the electron had a g factor of 2, the anomalous Zeeman 
effect could be explained, as well as the existence of multiplet splittings. 
There was a difficulty, however, in that the observed fine-structure intervals 
were only one-half the theoretically expected values. If a g factor of unity 
were chosen, the fine-structure intervals were given correctly, but the 
Zeeman effect was then the normal one. The complete explanation of spin, 
including correctly the g factor and the proper fine-structure interaction, 
came only with the relativistic electron theory of Dirac. But within the 



[Sect. 11.5] Special Theory of Relativity 365 

framework of an empirical spin angular momentum and a g factor of 2, 
Thomas showed that the origin of the discrepancy was a relativistic 
kinematic effect which, when included properly, gave both the anomalous 
Zeeman effect and the correct fine-structure splittings. The Thomas 
precession, as it is called, also gives a qualitative explanation for a spin- 
orbit interaction in atomic nuclei and shows why the doublets are 
"inverted" in nuclei. 

The Uhlenbeck-Goudsmit hypothesis was that an electron possessed a 
spin angular momentum S (which could take on quantized values of ±h/2 
along any axis) and a magnetic moment p related to S by 

\l = — S (11.39) 

mc 

The customary relation between magnetic moment and angular momentum 
is given by (5.64). Equation (1 1.39) shows that the electron has a g factor 
of 2. Suppose that an electron moves with a velocity v in external fields E 
and B. Then the equation of motion for its angular momentum in its rest 
frame is 

— = {x x B' (11.40) 

dt 

where B' is the magnetic induction in that frame. We will show in Section 
11.10 that in a coordinate system moving with the electron the magnetic 
induction is 

B'~ <B-- xEJ (11.41) 

where we have neglected terms of the order of (v 2 /c 2 ). Then (11.40) 
becomes 

— =|ix B--xE (11.42) 

dt \ c I 

Equation (11.42) is equivalent to an energy of interaction of the electron 
spin: 



r= -ji. IB-- x e) 



U' = -p. IB-- xEl (11.43) 

In an atom the electric force eE can be approximated as the negative 
gradient of a spherically symmetric average potential energy V(r). For one- 
electron atoms this is, of course, exact. Thus 

r dV 
eE=--— (11.44) 

r dr 



366 



Classical Electrodynamics 




Fig. 11.9 

Then the spin-interaction energy can be written 

U' = - — S • B + -k (S • L) - — 



(11.45) 



mc m"C r dr 

where L = m(r x v) is the electron's orbital angular momentum. This 
interaction energy gives the anomalous Zeeman effect correctly, but has a 
spin-orbit interaction which is twice too large. 

The error in (11.45) can be traced to the incorrectness of (11.40) as an 
equation of motion for the electron spin. The left-hand side of (11.40) 
gives the rate of change of spin in the rest frame of the electron. This is 
equal to the applied torque ({jl x B') only if the electron's rest frame is not 
a rotating coordinate system. If, as Thomas first pointed out, that co- 
ordinate system rotates, then the time rate of change of any vector G in that 
system is* 

dG = ldG\ 

dt \ dt / nonrot 

where <o T is the angular velocity of rotation found by Thomas, 
applied to the electron spin, (11.46) gives an equation of motion: 

dS =r (eB' 
dt 
The corresponding energy of interaction is 

U=U' -S-u> T (11.48) 

where U' is the electromagnetic spin interaction (11.45). 

The origin of the Thomas precessional frequency <a T is the acceleration 
experienced by the electron as it moves in its atomic orbit. Figure 11.9 
shows the electron at position 1 at a time t with instantaneous velocity 



ttirp X G 



I + Wyl 

\mc 1 



(11.46) 



When 



(11.47) 



See, for example, Goldstein, p. 133. 



[Sect. 11.5] 



Special Theory of Relativity 



367 



vector v, and at position 2 an infinitesimal time later (/ + bt) with 
velocity v + by. The increment in velocity is related to the electron's 
acceleration a by <5v = a dt. At time t the electron's rest frame K' and the 
laboratory frame K are related by a Lorentz transformation with velocity 
v. At time t + bt the electron's rest frame has now changed to K", 
related to K by a Lorentz transformation with velocity v + by. The 
question now arises, "How are the coordinate frames K" and K' related ? 
That is, how do the axes in the electron's rest frame behave in time ?" As 
viewed from the laboratory, in a time dt the electron undergoes an infini- 
tesimal change in velocity <5v. Consequently we might anticipate that K" 
and K' would be connected by a simple infinitesimal Lorentz transfor- 
mation. If so, (11.45) would be correct as it stands. To see that the con- 
nection is more than a mere Lorentz transformation we note that the 
transformation from K' to K" is equivalent to two successive Lorentz 
transformations, one with velocity — v, and the other with velocity v + by. 

K" 



K' -^-\\-+K— |v+ by 



(11.49) 

Now it is generally true that two successive Lorentz transformations are 
equivalent to a single Lorentz transformation plus a rotation. Using the 
general formula (11.21) twice, it is a straightforward matter to show that 
the time variables in K" and K' are related by 



t" = f - A 



l 



i- L , 

c 



by + 




- 1 



<5v 



(11.50) 



correct to first order in 6\. This shows that the direct transformation from 
K' to K" involves an infinitesimal Lorentz transformation with a velocity 



Av = 



1 



c 2 



<5v + 




The corresponding transformation of the coordinates is 



x » _ x - + /_i _ A x < x (UL*) 



— Avf 



(11.51) 



(11.52) 



Comparison with x" = x' + x' x A£l for a rotation of axes by an in- 
finitesimal angle ASl shows that the coordinate axes in K" are rotated 
relative to those in K' by an angle 

Aii = /-J_= - l\T-i£ (11.53) 






368 Classical Electrodynamics 

This shows that the coordinate axes in the electron's rest frame precess 
with an angular velocity 




where the result on the right is an approximation valid if v < c. We 
emphasize the purely kinematic origin of the Thomas precession by noting 
that nothing has been said about the cause of the acceleration. If a 
component of acceleration exists perpendicular to v, then there is a Thomas 
precession, independent of other effects such as precession of the magnetic 
moment in a magnetic field. 

For electrons in atoms the acceleration is caused by the screened 
Coulomb field (1 1.44). Thus the Thomas angular velocity is 

_ 1 r x vldV 1 T ldV 

to T ~ = L (11.55) 

2c 2 m r dr 2m 2 c 2 r dr 

It is evident from (11.48) and (11.45) that the extra contribution to the 
energy from the Thomas precession just reduces the spin-orbit coupling 
by a factor of \ (sometimes called the Thomas factor), yielding 

C /«_-Ls.B+-A 1 S.L±^ (11.56) 

mc 2m c r dr 

as the correct spin-orbit interaction energy for an atomic electron. 

In atomic nuclei the nucleons experience strong accelerations due to the 
specifically nuclear forces. The electromagnetic forces are comparatively 
weak. In an approximate way one can treat the nucleons as moving 
separately in a short-range, spherically symmetric, attractive, potential 
well, V N (r). Then each nucleon will experience in addition a spin-orbit 
interaction given by (11.48) with the negligible electromagnetic contri- 
bution U' omitted : 

[/jv^-S-Wj, (11.57) 

where the acceleration in to T is determined by V N {r). The form of w T is 
the same as (11.55) with V replaced by V N . Thus the nuclear spin-orbit 
interaction is approximately 

U N ~ ^- 9 S.L-^v (n.58) 

2M 2 c 2 rdr 

In comparing (11.58) with atomic formula (11.56) we note that both V 
and V N are attractive (although V N is much larger), so that the signs of 
the spin-orbit energies are opposite. This means that in nuclei the single 
particle levels form "inverted" doublets. With a reasonable form for V N , 



[Sect. 11.6] Special Theory of Relativity 369 

(11.58) is in qualitative agreement with the observed spin-orbit splittings 
in nuclei. 



11.6 Proper Time and the Light Cone 

In the previous sections we have explored some of the physical con- 
sequences of the special theory of relativity and Lorentz transformations. 
In the next two sections we want now to discuss some of the more formal 
aspects and to introduce some notation and concepts which are very useful 
in a systematic discussion of physical theories within the framework of 
special relativity. 

In Galilean relativity space and time coordinates are unconnected. 
Consequently under Galilean transformations the infinitesimal elements 
of distance and time are separately invariant. Thus 

ds 2 = dx 2 + dy 2 + dz 2 = ds' 2 



dt 2 = dt' 2 



(11.59) 



For Lorentz transformations, on the other hand, the time and space 
coordinates are interrelated. From (11.21) it is easy to show that the 
invariant "length" element is 

ds 2 = dx 2 + dy 2 + dz 2 - c 2 dt 2 (1 1.60) 

This leads immediately to the concept of a Lorentz invariant proper time. 
Consider a system, which for definiteness we will think of as a particle, 
moving with an instantaneous velocity v{f) relative to some coordinate 
system K. In the coordinate system K' where the particle is instantaneously 
at rest the space-time increments are dx' = dy' = dz' = 0, dt' = dr. Then 
the invariant length (11.60) is 

-c 2 dr 2 = dx 2 + dy 2 + dz 2 - c 2 dt 2 (1 1.61) 

In terms of the particle velocity \(t) this can be written 



- d 'J l ' I 



(11.62) 



Equation (11.62) shows the time-dilatation effect already discussed. But 
much more important, by the manner of its derivation (11.62) shows that 
the time t, called the proper time of the particle, is a Lorentz invariant 
quantity. This is of considerable importance later on when we wish to 
discuss various quantities and their time derivatives. If a quantity behaves 
in a certain way under Lorentz transformations, then its proper time 



370 



Classical Electrodynamics 




Fig. 11.10 World line of a system and the 
light cone. The unshaded interior of the cone 
represents the past and the future, while the 
shaded region outside the cone is called 
"elsewhere." A point inside (outside) the 
light cone is said to have a time-like (space- 
like) separation from the origin. 



derivative will behave in the same way because of the invariance of dr. 
But its ordinary time derivative will not have the same transformation 
properties. From (11.62) we see that a certain proper time interval 
(t 2 — r x ) will be seen in the system K as a time interval 

C T 2 dr 
= , (11.63) 



u = 



V\r) 



where t x and t 2 are the corresponding times in K. 

Another fruitful concept in special relativity is the idea of the light cone 
and "space-like" and "time-like" separations between two events. Con- 
sider Fig. 1 1 .10, in which the time axis (actually ct) is vertical and the space 
axes are perpendicular to it. For simplicity only one space dimension is 
shown. At t = a physical system, say a particle, is at the origin. Because 
the velocity of light is an upper bound on all velocities, the space-time 
domain can be divided into three regions by a "cone," called the light cone, 
whose surface is specified by x 2 + y 2 + z 2 = c 2 t 2 . Light signals emitted 
at / = from the origin would travel out the 45° lines in the figure. But 
any material system has a velocity less than c. Consequently as time goes 
on it would trace out a path, called its world line, inside the upper half- 
cone, e.g., the curve OB. Since the path of the system lies inside the upper 
half-cone for times t > 0, that region is called the future. Similarly the 
lower half-cone is called the past. The system may have reached O by a 
path such as AO lying inside the lower half-cone. The shaded region 
outside the light cone is called elsewhere. A system at O can never reach 
or come from a point in space-time in elsewhere. 

The division of space-time into the past-future region and the elsewhere 
region can be emphasized by considering the invariant separation between 
two events P x {x x , y x , z x , t x ) and P 2 (x 2 , y 2 -> z 2> h) i n space-time : 

*i2 2 = (*i - * 2 ) 2 + G/i - y 2 f + (*i - * 2 ) 2 - c\t x - t 2 ) 2 (1 1.64) 



[Sect. 11.7] Special Theory of Relativity 371 

For any two events P x and P 2 there are two possibilities : (1) s n 2 > 0, 
(2) s 1% 2 < 0. If 5 12 2 > 0, the events are said to have a space-like separation, 
because it is always possible to find a Lorentz transformation to a new 
coordinate system K' where (// — t 2 ') = and 

* 12 2 = « - x 2 'f + (<// - y 2 'f + &' - z 2 ') 2 > (1 1.65) 

That is, the two events are at different space points at the same instant of 
time. In terms of Fig. 11.10, one of the events is at the origin and the other 
lies in elsewhere. If s 12 2 < 0, the events are said to have a time-like separ- 
ation. Then a Lorentz transformation can be found which will make x{ 

= *2, Vi = Vz, z i = z 2> and 

V= -c\t{ - t 2 'f < (11.66) 

In the coordinate system K' the two events are at the" same space point, 
but are separated in time. In Fig. 11.10, one point is at the origin and the 
other is in the past or future. 

The division of the separation of two events in space-time into two 
classes — space-like separations or time-like separations — is a Lorentz 
invariant one. Two events with a space-like separation in one coordinate 
system have a space-like separation in all coordinate systems. This means 
that two such events cannot be causally connected. Since physical inter- 
actions propagate from one point to another with velocities no greater 
than that of light, only events with time-like separations can be causally 
related. An event at the origin in Fig. 11.10 can be influenced causally 
only by the events which occur in the past region of the light cone. 



11.7 Lorentz Transformations as Orthogonal Transformations in 
Four Dimensions 

The Lorentz transformation (11.19) and the more general form (11.21) 
are linear relations between the space-time coordinates (x, y, z, i) and 
(x', y', z', t'\ subject to the constraint, 

x 2 _|_ y 2 + 2 2 _ c 2,2 = x >2 + y >2 + z >2 _ fr'2 (H.67) 

This constraint is very reminiscent of the constraint involved in the 
rotation of coordinate axes in three space dimensions. In fact, if we intro- 
duce the four space-time coordinates, 

x ± = x, x 2 = y, x 3 = z, # 4 = ict (11.68) 

then the constraint becomes 

F 2 = x* + V + * 3 2 + x? (1 1 .69) 



372 Classical Electrodynamics 

is an invariant under Lorentz transformations. This is then exactly the 
requirement that Lorentz transformations are rotations in a four-dimen- 
sional Euclidean space or, more correctly, are orthogonal transformations 
in four dimensions. The Lorentz transformation (11.21) can be written 
in the general form : 

V=IVv, /* =1,2, 3, 4 (11.70) 

v = l 

where the coefficients a^ v are constants characteristic of the particular 
transformation. The invariance of R 2 (11.69) forces the transformation 
coefficients a^ to satisfy the orthogonality condition : 

4 
2 fl MV^A = ^vA (11-71) 

« = 1 
With (11.71) it is easy to show that the inverse transformation is 

4 

x n = 2 <«v„ (11.72) 

v = l 

and that . 

4 

2 a v M fl ^ = <5vA (11.73) 

H = l 

Furthermore, if we solve the four equations (11.70) for x^ in terms of xj 
and compare the solution to (11.72), we find that the determinant of the 
coefficients has the value unity : 

det |a„ v | = 1 (11.74) 

In general the determinant can be ±1, but the choice of the minus sign 
implies an inversion followed by a rotation. 

To give some substance to the above formalities we exhibit explicitly 
the transformation coefficients a^ v for a Lorentz transformation from 
system K to a system K' moving with a velocity v parallel to the z axis 

'l 

10 

^-) = I n J (H-75) 

y iyfi 

iO —iyfl y 
We have introduced the convenient abbreviations : 



c 
1 

y = 



i-4 

c 2 



(11.76) 



[Sect. 11.7] 



Special Theory of Relativity 



373 



Fig. 11.11 



Lorentz transformation as 
rotation of axes. 




With definitions (11.68) and (11.70) it is elementary to show that (11.75) 
yields exactly the Lorentz transformation (11.19). 

The formal representation of transformation (1 1.75) as a rotation of axes 
in the x 3 , x± plane (with a; 4 drawn as if it were real) can be accomplished 
simply. Figure 11.11 shows a rotation of the axes through an angle ip . The 
coordinates of the point P relative to the two sets of axes are related by 



x 3 = cos ip x 3 + sin ip x± 
xl = —sin ip x 3 + cos ip £ 4 



(11.77) 



Comparison of the coefficients in (11.77) with the transformation coeffi- 
cients in (11.75) shows that the angle ip is a complex angle whose tangent 
is 

tanip = ip (11.78) 

This result can be obtained directly from (11.77) without reference to 
(11.75) by noting that the origin x 3 = moves with a velocity v in the 
system K. That the angle ip is complex is emphasized by the fact that its 
cosine is greater than unity (cos ip = y > 1). Consequently the graphical 
representation of a Lorentz transformation as a rotation is merely a formal 
device. 

In spite of the formal nature of the x 3 , # 4 rotation diagram the pheno- 
mena of FitzGerald-Lorentz contraction and time dilatation can be 
displayed graphically. Figure 11.12 shows the length contraction on the 
right and time dilatation on the left. The distance L in the frame K' is 
observed in the frame AT as L, represented by the horizontal line at constant 
time in K. Because of the complex nature of the angle ip, L appears on the 
figure as larger than L , but mathematically the two lengths are related by 



or 



Lcos ip = L 

7 J 



(11.79) 



Classical Electrodynamics 




Fig. 11.12 Time dilatation and Fitz- 

Gerald-Lorentz contraction in terms 

of a rotation of space-time axes. 



in agreement with (11.22). Similarly the time intervals T in the frame K' 
are seen in the frame K as intervals T, where 

T = T cos y> = yT (1 1.80) 

in accord with (11.24). 

Sometimes a graphical display of Lorentz transformations is made 
using a real time variable x = ct, rather than ar 4 . This is called a 
Minkowski diagram and has the virtue of dealing with real quantities. It 
has the major disadvantage that the coordinate grids in the two frames K 
and K' must be scaled according to a rectangular hyperboloid law, as can 
be seen from (11.67). The interested reader may refer to Minkowski's 
paper in the collection, The Principle of Relativity, by Einstein et al., for a 
discussion of these diagrams. 



11.8 4-Vectors and Tensors; Covariance of the Equations of Physics 



The transformation law (11.70) for the coordinates x^ defines the trans- 
formation properties of vectors in the four-dimensional space-time (1 1.68). 
Any set of four quantities A^ which transform in the same way as x^ is 
called a 4-vector. Under the Lorentz transformation (a^) A^is transformed 



into A' where 



Ap — 2, a nv^\ 



(11.81) 



If a quantity <f> is unchanged under a Lorentz transformation, it is called 
a scalar or a Lorentz scalar. The four quantities formed by differentiation 
of a Lorentz scalar with respect to x^ transform as a 4-vector. This can be 
shown as follows. Consider 



d<f> _^d<f> dx v 
dx' J-* dx v dx' 

* v = l M 



(11.82) 



[Sect. 11.8] Special Theory of Relativity 375 

From (11.72) it is evident that 

|^7 = «„ (11-83) 

Consequently 

^-^%*4- (ii- 84 ) 

ox v 



dx' *-i 



as required for the transformation of a 4-vector. By similar means it is 
elementary to show that the 4-divergence of a 4-vector is Lorentz invariant : 

ydAl = y<^ (1L85) 

With A = dcfyjdx^ in this expression, we find that the four-dimensional 
Laplacian operator is a Lorentz invariant operator : 

^-ifMg-** (1L86) 

v=l V n=l * 

If D 2 operates on some other function, such as a 4-vector A^ the resulting 
quantity retains the transformation properties of the function operated 
on. The scalar product of two 4-vectors A^ and B^ is readily proved to be 
invariant : 

(i4'-B')s2^'B/ = (i4-B) (11-87) 

H = l 

Lorentz 4-vectors are tensors of the first rank in a four-dimensional 
space. Higher-ranks tensors are defined in an analogous way. A second- 
rank tensor T is a set of sixteen quantities which transforms according 
to the law: 

T; v = I a^T* (11.88) 

A, (7 = 1 

Higher-rank tensors are formed by the inclusion of more and more factors 
a v . A tensor of the «th rank is a set of 4 n quantities which have a transfor- 
mation law involving a product of n coefficients a^, in obvious generali- 
zation of (1 1 .88). Just as the scalar product of two 4-vectors has rank one 
less than the original quantities, so certain contracted quantities can be 
formed from higher-rank tensors. For example, the scalar product of a 
tensor of the second rank and a 4-vector transforms as a 4-vector: 

B; = I T^A\ = ZaJ2 T, V A V ) (11.89) 

v = l A = l \v = l / 

This and similar relations can be proved using the orthogonality con- 
ditions (11.71) and (11.73). 



376 Classical Electrodynamics 

The volume element in the four-dimensional space-time (11.68) will be 
defined as the real quantity 

d*x = dx x dx 2 dx 3 dx (11.90) 

where dx Q = (l/i) dx^ = d{ci). The transformation law of the volume 
element is 

d 4 x' = d«> X *> X 3> «*') fa (U 91) 

0\X^ x^, x 3 , X^) 

But the Jacobian in (11.91) is just the determinant of the a (11.74). 
Consequently the 4- volume element d*x is a Lorentz invariant quantity. 

The first principle of Einstein is that the laws of physics have the same 
form in different Lorentz frames. This means that the equations which we 
write down to describe the physical laws must be covariant in form. By 
covariant we mean that the equation can be written so that both sides have 
the same, well-defined, transformation properties under Lorentz transfor- 
mations. Thus physical equations must be relations between 4-vectors, or 
Lorentz scalars, or in general 4-tensors of the same rank. This is necessary 
in order that a relation valid in one coordinate frame will also hold in the 
same form in another. Consider, for example, the inhomogeneous pair of 
Maxwell's equations. It will be shown in the next section that these can be 
written in the relativistic form 

\SdF 4tt 

y-r 1 = -J^ ?* = 1,2, 3, 4 (11.92) 

v tt dx * c 

where J ^ is a suitable current 4- vector, and F^ is the field strength 4-tensor. 
Since the 4-divergence of a 4-tensor is a 4- vector, (11.92) is a relation 
between two 4-vectors. In another reference frame K', we expect the same 
physical laws to take the same form, 

tf^' = -^ (11-93) 

^-J ox a c 

Using transformation (11.81), we find that (11.93) can be expressed in 
terms of quantities in the original coordinate frame as 

i>(i:2:-T'')- 

fl = l v = l v 

This shows that, if (1 1.92) holds in the original frame of reference, then it 
holds in all equivalent Lorentz frames. If the two sides of (1 1.92) had not 
had the same Lorentz transformation properties, this would obviously 
not be true. 



[Sect. 11.9] Special Theory of Relativity 377 

To conclude these formal considerations we introduce some simplifying 
notation. In what follows : 

1. Greek indices will be summed from 1 to 4. 

2. Roman indices will represent spatial directions and will be summed 
from 1 to 3. 

3. 4- vectors will be denoted by A^ with (A lt A 2 , A 3 ) the components of 
a space vector A and A^ = iA . This correspondence will sometimes be 
written 

A„ = (A,iA ) (11.95) 

Sometimes the subscript on the 4-vector will be omitted, e.g. f(x) means 

/(x, 0- 

4. Scalar products of 4- vectors will be denoted by 

(A-B) = A- B- A B (11.96) 

where A • B is the ordinary 3-space scalar product. 

5. The summation convention will be used. That is, repeated indices 
are understood to be summed over, even though the summation sign is not 
written. If the repeated index is roman, the sum is from 1 to 3 ; if it is 
Greek, the sum is from 1 to 4. Thus, for example, (11.85) will be written 

dA v ' = BA, 

dx v ' dx^ 
and (11.89) will be written 

Tfi V A v = a llX T }v A v 

11.9 Covariance of Electrodynamics 

The invariance in form of the equations of electrodynamics under 
Lorentz transformations was shown by Lorentz and Poincare before 
Einstein formulated the special theory of relativity. We will now discuss 
this covariance and consider its consequences. There are two points of 
view possible. One is to take some experimentally proven fact such as the 
invariance of electric charge and try to deduce that the equations must be 
covariant. The other is to demand that the equations be covariant in form 
and to show that the transformation properties of the various physical 
quantities, such as field strengths and charge and current, can be satis- 
factorily chosen to accomplish this. Although the first view is to some the 
most satisfying, we will adopt the second course. Classical electrodynamics 
is correct, and it can be cast in covariant form. For simplicity we will 
consider the microscopic equations, without the derived quantities D 
andH. 



378 Classical Electrodynamics 

We begin with the continuity equation for charge and current densities : 

^=-V.J (11.97) 

dt 

This can be cast in covariant form by introducing the charge-current 4- 
vector Jp defined by 

J„ = (J, icp) (11.98) 

Then (11.97) takes on the obviously covariant form: 

^ = (11.99) 

That J^ is a legitimate 4-vector can be established from the experimentally 
known invariance of electric charge. This invariance implies that 
(p dx x dx 2 dx z ) is a Lorentz invariant. Since i d*x = (dx x dx 2 dx 3 dx^ is a 
Lorentz invariant, it follows that p transforms like the fourth component of 
a 4-vector. The transformation properties of J follow similarly. 

The wave equations for the vector potential A and the scalar potential 
O are 



c 2 dt 2 c 

V 2 0--— = -4tt P 



with the Lorentz condition 



(11.100) 



V-A + i— = (11.101) 

c dt 

The differential operator on the left-hand sides of the wave equations can 
be recognized as the Lorentz invariant four-dimensional Laplacian (1 1.86). 
The right-hand sides of these equations are the components of a 4-vector. 
Consequently, the requirement of covariance means that the vector and 
scalar potentials are the space and time parts of a 4-vector potential A : 

4, = (A,iO) (11.102) 

Then the wave equations can be written 

□ 2 ^ = - ^L j ju = 1,2, 3, 4 (11.103) 

c 

while the Lorentz condition becomes 

^ = (11.104) 

dx^ 



[Sect. 11.9] 



Special Theory of Relativity 



379 



We are now ready to consider the field strengths E and B. They are 
defined in terms of the potentials by 

c dt 
B = V x A J 

By writing out the components explicitly, for example, 



(11.105) 



dA x dA± 

ltl = a a 

ox± ox 1 

B = dA 3 dA 2 



(11.106) 



it is evident that the electric field and the magnetic induction are elements 
of the second-rank, antisymmetric, field-strength tensor F^: 

(11.107) 



P _2±._ 


dA_, 
dx v 




Explicitly, the field-strength tensor is 






/ B z 


-B 2 


— IE^ 


l-B 3 


*i 


— iE 2 


(iV) = 1 B ^ _ Bi 





-iE z 


\ iE x iE% 


iE 3 






(11.108) 



To complete the demonstration of the covariance of electrodynamics 
we must consider Maxwell's equations. The inhomogeneous pair are 



V • E = 4-rrp 

c dt c 



(11.109) 



Since the right-hand sides form the components of a 4- vector, so must the 
left-hand sides. With definition (11.108) of the field-strength tensor it is 
easy to show that the left-hand sides in (11.109) are the divergence of the 
field-strength tensor. Thus (11.109) takes the covariant form 



wl JIV ^" r 

dx v c 

Similarly the two homogeneous Maxwell's equations, 

1 f5R 

V • B = 0, VxE + ~=0 

c dt 



(11.110) 



(11.111) 



380 Classical Electrodynamics 

can be shown to reduce to the four equations : 

dF„ v dF Xll dF vX 

— ^ + -^ + -^ = (11.112) 

ox x dx v dXp 

where A, ju, v are any three of the integers 1 , 2, 3, 4. Each term in (1 1 . 1 12) 
transforms like a 4-tensor of the third rank so that the equation is covariant 
in form, as required. 



11.10 Transformation of the Electromagnetic Fields 

Since the fields E and B are elements of the field-strength tensor F , 
their transformation properties can be found from 

F'.y = a^a^F^ (11.113) 

With transformation (11.75) from a system Kto K' moving with velocity 
v along the x 3 axis, (11.113) gives the transformed fields: 

E{ = 7(E X - PB 2 ) 2?/ = y{B 1 + 0EJ 
E* = y(E 2 + PBJ B 2 ' = y(B 2 - $E X ) 



Ez — E 3 B 3 = B 3 



(11.114) 



The inverse transformation can be obtained from (11.114) by the inter- 
change of primed and unprimed quantities and (5 -> — /?. For a general 
Lorentz transformation from K to a system K' moving with velocity v 
relative to K, the transformation of the fields is evidently 

E M ' = E M B„ ' = B M 

E/ = y( El + - c x b) B/ = y ( Bl - 1 x E ) J (1L115) 

Here || and j^ mean parallel and perpendicular to the velocity v. Transfor- 
mation (11.115) shows that E and B have no independent existence. A 
purely electric or magnetic field in one coordinate system will appear as a 
mixture of electric and magnetic fields in another coordinate frame. Of 
course certain restrictions apply (see Problem 11.10) so that, for example, 
a purely electrostatic field in one coordinate system cannot be transformed 
into a purely magnetostatic field in another. But the fields are completely 
interrelated, and one should properly speak of the electromagnetic field 
F^, rather than E or B separately. 

As an example of the transformation of the electromagnetic fields we 
consider the fields seen by an observer in the system K when a point charge 
q moves by in a straight-line path with a velocity v. The charge is at rest 
in the system K', and the transformation of the fields is given by the inverse 



[Sect. 11.10] 



Special Theory of Relativity 



381 



Fig. 11.13 Particle of charge q 

moving at constant velocity v 

passes an observation point P 

at impact parameter b. 



X\ 


^\ r 


X\ 








a v 

i > — 


X3 






9 


*3 



*2 



'X% 



of (1 1.1 14) or (11.115). We suppose that the charge moves in the positive 
x 3 direction and that its closest distance of approach to the observer is b. 
Figure 11.13 shows a suitably chosen set of axes. The observer is at the 
point P. At / = t' = the origins of the two coordinate systems coincide 
and the charge q is at its closest distance to the observer. In the frame K' 
the observer's point P, where the fields are to be evaluated , has coord inates 
Xl ' = b, x 2 ' = 0, < = — vt', and is a distance r' = Vb 2 + {vt'f away 
from*?. We will need to express r' in terms of the coordinates of K. The 
only coordinate needing transformation is the time t ' = y[t — (vjc 2 )x 3 ] = 
yt, since x 3 = for the point P in the frame K. In the rest frame K' of the 
charge the electric and magnetic fields are 



£/ = 0, 



£,' = 0, 



B 2 ' = 0, 



E,' = 



BJ = 



qvt' 

~'3 



(11.116) 



In terms of the coordinates of K the nonvanishing field components are 



£/ = 



qb 



£,'= - 



qyvt 



(11.117) 



(6 2 + yW? A ' ~* (b 2 + y W)° A 

Then, using the inverse of (11.114), we find the transformed fields in the 
system K: 

yqb 



E x = yE{ = 



£o = £q = — 



(&*.+ yW) 8/ * 

qyvt 



(b 2 + yWf* 
B 2 = y^ = fiEi 



(11.118) 



with the other components vanishing. 

Fields (11.118) exhibit interesting behavior when the velocity of the 
charge approaches that of light. First of all there is observed a magnetic 



382 



Classical Electrodynamics 



induction in the x 2 direction. This magnetic field becomes almost equal to 
the transverse electric field E x as /8 -> 1. Even at nonrelativistic velocities 
where y c=L 1, this magnetic induction is equivalent to 



B 



q v x r 



(11.119) 



which is just the Ampere-Biot-Savart expression for the magnetic field of 
a moving charge. This can obviously be obtained directly from the 
inverse of (11.115). At high speeds when y > 1 we see that the peak 
transverse electric field E x (t = 0) becomes equal to y times its nonrelati- 
vistic value. In the same limit, however, the duration of appreciable field 
strengths at the point P is decreased. A measure of the time interval over 
which the fields are appreciable is evidently 



Lt~± 



b_ 

yv 



(11.120) 



As y increases, the peak fields increase in proportion, but their duration 
goes in inverse proportion. The time integral of the fields times v is 
independent of velocity. Figure 11.14 shows this behavior of the transverse 
electric and magnetic fields and the longitudinal electric field. For /S -»- 1 
the observer at P sees nearly equal transverse and mutually perpendicular 
electric and magnetic fields. These are indistinguishable from the fields 
of a pulse of plane polarized radiation propagating in the x z direction. 









79 
b 2 






P<1- 








-W 


L_it-I 




Is" 


***v\ 









vt >- 




Fig. 11.14 Fields due to a uniformly moving, charged particle as a function of time. 



[Sect. 11.11] Special Theory of Relativity 383 

The extra longitudinal electric field varies rapidly from positive to 
negative and has zero time integral. If the observer's detecting apparatus 
has any significant inertia, it will not respond to this longitudinal field. 
Consequently for practical purposes he will see only the transverse fields. 
This equivalence of the fields of a relativistic charged particle and those of 
a pulse of electromagnetic radiation will be exploited later in Chapter 1 5. 
That a plane electromagnetic wave in one coordinate frame K will also 
appear as a plane wave in another coordinate frame K' moving with 
constant velocity relative to K follows from the invariant form of the wave 
equation under Lorentz transformations. Thus in the frame K a plane 
wave is represented by 

^v(x,0=/,/ k - x -^ (11-121) 

where f^ are appropriate constant coefficients, and k and co are the wave 
vector and frequency of the wave. In the coordinates system K' the plane 
will be 

i 7 ;v(x',o=/;v^ k '- x '~ icor (n.122) 

where the f'^ are again constant coefficients, and k' and co' are the wave 
vector and frequency as seen in K'. According to (1 1.1 13), the two sets of 
fields are related by 

ri ik' .x' —i<o't' r ik.x—ia>t /i i n ^ 

//•v e = a&<*v*fjL*e (11.123) 

In order that (11.123) be true at all points in space-time the phase factors 
on both sides must be equal: 

k'-x' — co't' = k.x - cot (11.124) 

This invariance of the phase means that k and co must form the space and 
time parts of a 4- vector k^: 



^('•t) 



(11.125) 



Then the invariance of phase becomes the obvious invariance of a scalar 
product (k • x) of two 4-vectors. The relativistic formulas for the Doppler 
shift follow immediately from (11.125), as was shown in Section 11.4. 



11.11 Covariance of the Force Equation and the Conservation Laws 

In Section 11.9 the covariance of electrodynamics was discussed from 
the point of view of charge and current densities and the resulting fields 
and potentials. We know that the sources of charge and current are 
ultimately charged particles which can move under the action of fields. 



384 Classical Electrodynamics 

Consequently to complete our discussion we must consider the covariant 
formulation of the Lorentz force equation and the conservation laws of 
momentum and energy. 

The Lorentz force equation can be written as a force per unit volume 
(representing the rate of change of mechanical momentum of the sources 
per unit volume) : 

f=pE + -J xB (11.126) 

c 

where J and p are the current and charge densities. Writing out a single 
component of f, we find 

A = P E X + - (J 2 B 3 - J 3 B 2 ) = - (F 12 J 2 + F 13 J 3 + F 14 J 4 ) (11.127) 
c c 

where we have used definitions (1 1 .98) and (1 1 .108). The other components 
of f yield similar results, showing that (11.126) can be written as 

f k = -F kv J v , k= 1,2,3 (11.128) 

c 

The right-hand side of (11.128) is evidently the space components of a 

4- vector. Hence f must be the space part of a 4- vector /„ = If, i — I, 
where : 

/„ = -V* ( 11129 > 

c 

To see the meaning of the fourth component of the force-density 4-vector 
we write out 

f = ~h = - {F^i + V2 + iVs) = E • J (11.130) 

i i 

But (E • J) is just the rate at which the field does work on the sources per 
unit volume, or the rate of change of mechanical energy of the sources per 
unit volume. Thus we see that the covariant form (1 1.129) of the Lorentz 
force equation gives the rate of change of mechanical momentum per unit 
volume as its space part, and the rate of change of mechanical energy per 
unit volume as its time part. Alternatively, it may be viewed as giving the 
space and time derivatives of something of the dimensions of work per 
unit volume. 

The conservation laws for mechanical plus electromagnetic energy and 
momentum derived in Chapter 6 can be presented in covariant form as 
the space and time components of a single 4-vector equation. If the in- 
homogeneous Maxwell's equations (11.110) are used to eliminate J v in 



[Sect. 11.11] Special Theory of Relativity 

(11.129), the force density becomes 

F„ 



J II . -* /JV 

47T 



1 - 9F V , 



dx } 



385 



(11.131) 



The right-hand side of (1 1.131) can be written as the divergence of a tensor 
of the second rank. We define the symmetric tensor T^, called the 
electromagnetic stress-energy-momentum tensor, 



T --L 

<terL 



FnxFxv + i^u V Fi a F_ 



fiX* Av 



? u ttv r A<r r A<r 



(11.132) 



It will be left to the problems (Problem 11.12) to show that by means of 
the homogeneous Maxwell's equations and (1 1.132) force equation (11.131) 
can be written in the form: 

/„ = ^ (H-133) 

ox v 

The tensor T^ v can be written out explicitly in terms of the fields using 
(11.132): 

'll -1-19 -lift 



<T„) = 



J 21 



T 2 2 T 23 

^32 -*33 

-icg 2 -icg 3 



-Kgl) 

-icgz 

-icgz 

u 



(11.134) 



where T ik is the symmetric Maxwell's stress tensor defined on page 194, 
g is the electromagnetic momentum density, 



g = — E xB = ^S 
Aire c 2 



and u is the energy density, 



u = — (E 2 + B 2 ) 



(11.135) 



From definition (6.102) of the spatial parts of T^ [or from (11.132)], we 
see that the stress-energy-momentum tensor has a vanishing trace : 

IT,, = (11.136) 

The conservation laws of momentum and energy are merely the three- 
dimensional integrals of the force equation (1 1.133). To see this we write 
out a typical spatial-component equation : 



Jk — 



dr t 



dx„ 



dx, dx. 



v A kv w l ki i w *M /T7 . "T \ "gk 



dt 



(11.137) 



386 Classical Electrodynamics 

If we identify the spatial integral of f k as the rate of change of the kth 
component of mechanical momentum P k , then the integral of (11.137) can 
be written 

4 (P + G) = I V •¥ d z x = i> n V da (11.138) 

dt Jv Js 

where G k is the kth component of total electromagnetic momentum. This 
is the momentum-conservation law already obtained in Chapter 6. 
Similarly the fourth component of (11.133) can be written 

/ = £/ 4 = E .J = -^ + -^= -V-S-^ (11.139) 
i i ox i i ox± dt 

With the volume integral of f identified as the rate of change of total 
mechanical energy T, the conservation of energy law is 

-{T+ U)= - I V-Sd 3 x = - <t> n-S da (11.140) 

dt Jv Js 

where U is the total electromagnetic energy in the volume V. 



REFERENCES AND SUGGESTED READING 

The theory of relativity has an extensive literature all its own. To my mind the most 
lucid, though concise, presentation of special and general relativity is the famous 1921 
article (recently brought up to date) by 

Pauli. 
In addition, there are a number of textbooks devoted to special relativity at the graduate 
level, some of which are 

Aharoni, 

Bergmann, Chapters I-IX, 

Moller, Chapters I-VII. 
Moller's book is perhaps the most authoritative. 

The flavor of the original theoretical developments can be obtained by consulting 
the collected papers of 

Einstein, Lorentz, Minkowski, and Weyl. 
The main experiments are summarized briefly, but clearly, in 

Moller, Chapter I, 

Panofsky and Phillips, Chapter 14. 
A fuller description of the experimental basis of special relativity, with many references, 
is presented in 

Condon and Odishaw, Part 6, Chapter 8 by E. L. Hill. 
Thomas precession is discussed by 

Moller, Sections 22 and 47, 
in a manner akin to ours. A different approach to the problem is given by 

Corben and Stehle, Section 92. 



[Probs. 11] 



Special Theory of Relativity 



387 



PROBLEMS 

11.1 A possible clock is shown in the figure. It consists of a flashtube F and a 
photocell P shielded so that each views only the mirror M, located a 
distance d away, and mounted rigidly with respect to the flashtube-photo- 
cell assembly. The electronic innards of the box are such that, when the 
photocell responds to a light flash from the mirror, the flashtube is 
triggered with a negligible delay and emits a short flash towards the 
mirror. The clock thus "ticks" once every (2d/c) seconds when at rest. 




11.2 



(a) Suppose that the clock moves with a uniform velocity v, perpen- 
dicular to the line from PF to M, relative to an observer. Using the 
second postulate of relativity, show by explicit geometrical or algebraic 
construction that the observer sees the relativistic time dilatation as the 
clock moves by. 

(Z>) Suppose that the clock moves with a velocity v parallel to the line 
from PF to M. Verify that here, too, the clock is observed to tick more 
slowly, by the same time dilatation factor. 

(a) Show explicitly that two successive Lorentz transformations in the 
same direction commute and that they are equivalent to a single Lorentz 
transformation with a velocity 

v x + v 2 



1 + Ov 2 /c 2 ) 



11.3 



This is an alternative way to derive the parallel-velocity addition law. 

(b) Show explicitly that two successive Lorentz transformations at right 
angles (y x in the x direction, v % in the y direction) do not commute. Show 
further that in whatever order they are applied the result is not the same 
as a single transformation with y =w x + jv 2 . Give one or more simple 
reasons why this result is necessary within the framework of special 
relativity. 

(a) Find the form of the wave equation in system A" if it has its standard 



388 Classical Electrodynamics 

form in system K' and the two coordinate systems are related by the 
Galilean transformation x' = x — vt, t' = t. 

(b) Show explicitly that the form of the wave equation is the same in 
system K as in K' if the coordinates are related by the Lorentz trans- 
formation x' = y(x — vt), t' = y[t — (vx/c 2 )]. 
11.4 A coordinate system K' moves with a velocity v relative to another system 
K. In K' a particle has a velocity u' and an acceleration a'. Find the 
Lorentz transformation law for accelerations, and show that in the system 
K the components of acceleration parallel and perpendicular to v are 



9,,= FW a " 

ll-t\ 

\ c 2 } ( v \ 

a ±=7 X3 W + - 2 X < a ' Xu) 



11.5 Assume that a rocket ship leaves the earth in the year 2000. One of a set 
of twins born in 1980 remains on earth; the other rides in the rocket. 
The rocket ship is so constructed that it has an acceleration g in its own 
rest frame (this makes the occupants feel at home). It accelerates in a 
straight-line path for 5 years (by its own clocks), decelerates at the same 
rate for 5 more years, turns around, accelerates for 5 years, decelerates for 
5 years, and lands on earth. The twin in the rocket is 40 years old. 

(a) What year is it on earth? 

(b) How far away from the earth did the rocket ship travel ? 

11.6 In the reference frame K two very evenly matched sprinters are lined up a 
distance d apart on the y axis for a race parallel to the x axis. Two 
starters, one beside each man, will fire their starting pistols at slightly 
different times, giving a handicap to the better of the two runners. The 
time difference in K is T. 

(a) For what range of time differences will there be a reference frame K' 
in which there is no handicap, and for what range of time differences is 
there a frame K' in which there is a true (not apparent) handicap? 

(b) Determine explicitly the Lorentz transformation to the frame K' 
appropriate for each of the two possibilities in (a), finding the velocity of 
K' relative to K and the space-time positions of each sprinter in K'. 

11.7 Using the four-dimensional form of Green's theorem, solve the inhomo- 
geneous wave equations 

— 4n 

(a) Show that for a localized charge-current distribution the 4-vector 
potential is 

A(x) = I fep d^ 

" 7TC J R 2 



[Probs. 11] Special Theory of Relativity 389 

where R 2 = (x — f) • (x — |), x means (x lt x 2 ,x 3 , # 4 ), and J 4 | = d£ x d£ 2 d£ 3 d| 4 . 
(b) From the definitions of the field strengths F^ show that 



= _2 (*(/ x 

TTCj R 



/*" -~ I — 7j3 — s 



where (7 x i?)^ = /^/f, - J V R^ 

11.8 The three-dimensional formulation of the radiation problem leads to the 
retarded solution 



A^t)^lj J J^l2 



(PS 

t'=t-(rje) 



where r = |x — %\. Show the connection between this retarded solution 
and the solution of Problem 11.7 by explicitly performing the integration 
over d£ 4 . 

11.9 A classical point magnetic moment f* at rest has a vector potential 



and no scalar potential. Show that, if the magnetic moment moves with a 

velocity v(v <^ c), there is an electric dipole moment p associated with the 

magnetic moment, where 

v 
p =- X {X 

c 

What can you say if v is not small in magnitude compared toe? Show 
that the interaction energy between the moving dipole and fields E and B 
is the same as would be obtained by calculating the magnetic field in the 
rest frame of the magnetic moment. 

11.10 (a) Show that (B 2 — E 2 ) is an invariant quantity under Lorentz trans- 
formations. What is its form in four-dimensional notation ? 
(b) The symbol e Xflva is defined to have the properties 



ro if 



if any two indices are equal 

for an even (odd) permutation of indices 



€ Xllva is a completely antisymmetric unit tensor of the fourth rank (actually 
a pseudotensor under spatial inversion). Prove that ^XfivaF^va (sum- 
mation convention implied) is a Lorentz invariant, and find its form in 
terms of E and B. 

11.11 In a certain reference frame a static, uniform, electric field E is parallel to 
the x axis, and a static, uniform, magnetic induction B = 2E lies in the 
x-y plane, making an angle 6 with the x axis. Determine the relative 
velocity of a reference frame in which the electric and magnetic fields are 
parallel. What are the fields in that frame for << 1 and -»> (tt/2) ? 



390 Classical Electrodynamics 

11.12 Show that the force equation/,, = (l/c)F MV J v can be written as 

dT 

U dx v 
where 

T l*v = -^\. F nX F X V + i S nv F XaFxo\ 

11.13 A pulse of electromagnetic radiation of finite spatial extent exists in charge- 
and current-free space. 

(a) By means of the divergence theorem in four dimensions, prove that 
the total electromagnetic momentum and energy transform like a 4- vector. 

(b) Show that for a plane wave this 4- vector has zero "length," but 
that for other possible field configurations (e.g., spherically diverging wave) 
this is not true. 



12 



Relativistic-Particle 
Kinematics and Dynamics 



In Chapter 1 1 the special theory of relativity was developed with 
particular emphasis on the electromagnetic fields and the covariance of 
the equations of electrodynamics. Only in Section 11.11 was there a 
mention of the mechanical origin of the sources of charge and current 
density. The emphasis on electromagnetic fields is fully justified in the 
presentation of the first aspects of relativity, since it was the behavior of 
light which provided the puzzling phenomena that were understood in 
terms of the special theory of relativity. Furthermore, a large class of 
problems can be handled without inquiry into the detailed mechanical 
behavior of the sources of charge and current. Nevertheless, problems 
which emphasize the fields rather than the sources form only a part of 
electrodynamic phenomena. There is the converse type of problem in 
which we are interested in the behavior of charged particles under the 
action of applied electromagnetic fields. The particles represent charge 
and current densities, of course, and so act as sources of new fields. But 
for most applications these fields can be neglected or taken into account in 
an approximate way. In the present chapter we wish to explore the motion 
of relativistic particles, first their kinematics and then their dynamics in 
external fields. Discussion of the difficult problem of charged particles 
acting as the sources of fields and being acted on by those same fields will 
be deferred to Chapter 17. 



12.1 Momentum and Energy of a Particle 

In nonrelativistic mechanics a particle of mass m and velocity v has a 
momentum p = rav and a kinetic energy T = \mv^. Newton's equation 

391 



392 Classical Electrodynamics 

of motion relates the time rate of change of momentum to the applied 
force. For a charged particle the force is the Lorentz force. Since we have 
discussed the Lorentz transformation properties of the Lorentz force density 
in Section 11.11, we can immediately deduce the behavior of a charged 
particle's momentum under Lorentz transformations. For neutral particles 
with no detectable electromagnetic interactions it is clearly impossible to 
obtain their relativistic transformation properties in this way, but there 
is ample experimental evidence that all particles behave kinematically in 
the same way, whether charged or neutral. 

A charged particle can be thought of as a very localized distribution of 
charge and mass. To find the force acting on such a particle we integrate 
the Lorentz force density/^ (11.129) over the volume of the charge. If the 
total charge is e and the velocity of the particle is v, then the volume 
integral of (11. 129) is 

f tl d 3 x = e -F MV v v (12.1) 

c 



S 



where v„ = (v, ic), and F^ v is interpreted as the average field acting on the 
particle. The left-hand side of (12.1) is now to be equated to the time rate 
of change of the momentum and energy of the particle, just as in Section 
11.11. Thus 

^ -//„«*•* (12.2) 

where we have written p k as the kXh component of the particle's momentum 
and /> 4 = iEjc as proportional to the particle's energy. That p^ is indeed 
a 4- vector follows immediately from (12.2). If we integrate both sides with 
respect to time, then the left-hand side becomes the momentum or energy 
of the particle while the right-hand side is the four-dimensional integral 
of f^. Since d*x is a Lorentz invariant quantity, it follows that p^ must 
have the same transformation properties as/„. Therefore the momentum 
p and the energy E of a particle form a 4- vector p^ : 

Pll =U-) (12-3) 



" v c 



The transformation of momentum and energy from one Lorentz frame K 
to another K' moving with a velocity v parallel to the z axis is 



Pi = Pi> Pz = Pz 

p s = y \Ps + P —J 

E = y(E' + jfcfl,') 



(12.4) 



[Sect. 12.1] Relativistic-F 'article Kinematics and Dynamics 393 

where /? = vjc and y = (1 — /S 2 ) - ^. The inverse transformation is 
obtained by changing /5 — »► — /5 and interchanging the primed and unprimed 
variables. 

The length of the 4- vector p^ is a Lorentz invariant quantity which is 
characteristic of the particle : 

(P'P) = (p'-p')=-- 2 (12.5) 

c 2 

In the rest frame of the particle (p' = 0) the scalar product (12.5) gives the 
energy of the particle at rest : 

E' = X (12.6) 

To determine X we consider the Lorentz transformation (12.4) of p^ from 
the rest frame of the particle to the frame Kin which the particle is moving 
in the z direction with a velocity v. Then the momentum and energy are 



-*-># 



(12.7) 



E = yX 

From the nonrelativistic expression for momentum p = my we find that 
the invariant constant X = mc 2 . The nonrelativistic limit of the energy is 

E = ymc 2 ~ mc 2 + \mv 2 -\ (12.8) 

This shows that is is the total energy of the particle, consisting of two parts : 
the rest energy (mc 2 ) and the kinetic energy. Even for a relativistic particle 
we can speak of the kinetic energy T, defined as the difference between the 
total and the rest energies : 

T = E - mc 2 = (y - \)mc 2 (12.9) 

In summary, a free particle with mass m moving with a velocity v in a 
reference frame K has a momentum and energy in that frame : 

p = yms ) 

E = ymc* J 

From (12.5) it is evident that the energy E can be expressed in terms of the 
momentum as 

E = (c 2 p 2 + m 2 c*) v * (12.11) 

The velocity of the particle can likewise be expressed in terms of its 
momentum and energy : 

v = ^ (12.12) 

E 



394 



Classical Electrodynamics 



In dealing with relativistic-particle kinematics it is convenient to adopt 
a consistent, simple notation and set of units in which to express momenta 
and energies. In the formulas above we see that the velocity of light appears 
often. To suppress various powers of c and so simplify the notation we 
will adopt the convention that all momenta, energies, and masses will be 
measured in energy units, while velocities are measured in units of the 
velocity of light. All powers of c will be suppressed. Consequently in 
what follows, the symbols 



P 
E 

m 

v 



stand for 



cp 
E 

v/c 



(12.13) 



As energy units, the ev (electron volt), the Mev (million electron volt), and 
the Bev (10 9 ev) are convenient. One electron volt is the energy gained by 
a particle with electronic charge when it falls through a potential difference 
of one volt (1 ev = 1.602 x 10" 12 erg). 



12.2 Kinematics of Decay Products of an Unstable Particle 

As a first illustration of relativistic kinematics which follow immediately 
from the 4- vector character of the momentum and energy of particles, we 
consider the two-body decay of an unstable particle at rest. Such decay 
processes are common among the unstable particles. Some examples are 
the following. 

1. Charged pi meson decays into a mu meson and a neutrino with a 
lifetime t = 2.6 x 10~ 8 sec: 

77" — *■ jU + V 

The pi-meson rest energy is M = 139.6 Mev, while that of the mu meson 
is m^ = 105.7 Mev. The neutrino has zero rest mass, m v = 0. There is, 
therefore, an energy release of 33.9 Mev in pi-meson decay. 

2. Charged K meson sometimes decays into two pi mesons with a 
lifetime t = 1.2 x 10 -8 sec: 



K± 



+ 77° 



The charged K meson has a rest energy M = 494 Mev, while the two pi 
mesons have rest energies, m ± = 139.6 Mev, m = 135.0 Mev. Thus the 
energy release is 219 Mev. 



[Sect. 12.2] Relativistic-Particle Kinematics and Dynamics 395 

3. Lambda hyperon decays into a neutron or a proton and a pi meson 
with a lifetime t = 2.9 x 10 -10 sec: 

\n + 77° 

The rest energy of the lambda hyperon is M = 1115 Mev; that of the 
proton m p = 938.5 Mev, and of the neutron m n = 939.8 Mev. With the 
pi-meson masses given above, we find that the energy release in lambda 
decay is 37 Mev in the charged mode and 40 Mev in the neutral mode. 

The transformation of a system of mass M at rest into two particles of 
mass m 1 and m 2 

M^m 1 + m 2 (12.14) 

can occur if the initial mass is greater than the sum of the final masses. 
We define the mass excess AM: 

AM=M-m 1 -m 2 (12.15) 

The sum of the kinetic energies of the two particles must be equal to AM. 
Since the initial system had zero momentum, the two particles must have 
equal and opposite momenta, p x = — p 2 = p. From (12.11) the conser- 
vation of energy can be written 

Vp 2 + m 2 + Vp 2 + m 2 = M (12.16) 

From this equation it is a straightforward matter to find the magnitude of 
the momentum p and the individual particle energies, E 1 and E 2 . 

Rather than solve (12.16) we wish to obtain our answers by illustrating 
a useful technique which exploits the Lorentz invariance of the scalar 
product of two 4- vectors. The conservation of energy and momentum in 
the two-body decay can be written as a 4- vector equation : 

P=Pi+P2 (12.17) 

where the 4-vector subscript ju on each symbol has been suppressed. The 
squares of the 4-vector momenta are the invariants : 

(P ■ P) = - M \ ( Pl -pj = -m 2 , (p 2 -p 2 ) = -m 2 (12.18) 

In (12.18) we have written the squares of the 4-vectors as self-scalar 
products in order to distinguish the square of a vectorial quantity as a 
three-space self-scalar product (e.g., p 2 = p • p). Using (12.17), we form 
the square of the 4-vector p 2 : 

(Pz-Pz) = (P-Pi)-(P-p 1 ) } 

(12.19) 
or -m 2 = -M 2 - m 2 - 2(P- Pl ) J 



M 2 + m 2 - 


2 

m 2 - 


2M 




M 2 + m 2 - 


m 2 



396 Classical Electrodynamics 

The scalar product (P • p x ) is Lorentz invariant. In the frame in which the 
system M is at rest its space part vanishes, and it has the value : 

(P-p 1 ) = -ME X (12.20) 

Therefore the total energy of the particle with mass m x is 

E x = M " + m ± ~ m2 (12.21) 

Similarly 

E 2 = "' ' "'* '" 1 (12.22) 

2M 

Often it is more convenient to have expressions for the kinetic energies 
than for the total energies. Using (12.15), it is easy to show that 

Ti = AM (l-^-^), i_l f 2 (12.23) 

where AM is the mass excess. The term AM/2M is a relativistic correction 
absent in the nonrelativistic result. Although it may not have obvious 
relativistic origin, a moment's thought shows that, if AM/2M is appreciable 
compared to unity, then necessarily the outgoing particles must be treated 
relativistically. 

As a numerical illustration we consider the first example listed above, 
the decay of the pi meson. The mass excess is 33.9 Mev, while M = 139.6 
Mev, m^ = 105.7 Mev, m v = 0. Consequently the mu-meson and neutrino 
kinetic energies are 

Tf „ 33.9(l -i2LZ --£?-) = 4.1 Mev 
M \ 139.6 2(139.6)/ 

T v = 33.9 - 7; = 29.8 Mev 

The unique energy of 4. 1 Mev for the mu meson was the characteristic of 
pi-meson decay at rest which led to its discovery in 1947 by Powell and 
coworkers from observations in photographic emulsions. 

The lambda particle was first observed in flight by its charged decay 
products (p + 77~) in cloud chambers. The charged particle tracks appear 
as shown in Fig. 12.1. The particles' initial momenta and identities can 
be inferred from their ranges and their curvatures in a magnetic field (or 
by other techniques, such as grain counting, in emulsions). The opening 
angle 6 between the tracks provides the other datum required to determine 
the unseen particle's mass. Consider the square of (12.17): 

(P'P) = (p 1 +p 2 )'(p 1 +p 2 ) (12.24) 

This becomes 

-M 2 = -m 2 - m 2 + 2( Pl ■ p 2 ) (12.25) 



[Sect. 12.3] Relativistic-Particle Kinematics and Dynamics 397 



Fig. 12.1 Decay of lambda particle in flight. 

If the scalar product (p x ■ p^) is evaluated in the laboratory frame, we find 

M 2 = m? + m 2 2 + 2E X E 2 - 2p x p 2 cos (12.26) 

where p 1 and/? 2 are the magnitudes of the three-dimensional momenta. 

In a three- or more body decay process the particles do not have unique 
momenta, but are distributed in energy in some way. These energy spectra 
have definite upper end points which can be determined from the kine- 
matics in ways similar to those used here (see Problem 12.2). 

12.3 Center of Momentum Transformation and Reaction Thresholds 

A common problem in nuclear or high-energy physics is the collision 
of two particles. Particle 1 (the projectile), with mass m x , momentum 
p x = p, and energy E lt is incident on particle 2 (the target) of mass m 2 at 
rest in the laboratory. The collision may involve elastic scattering, 

® + (D^© + ® 02.27) 

where the primes mean that the directions of the particles are in general 
different. The collision may, on the other hand, be a reaction 

® + (D-(D + (!) + ••• (12.28) 

in which two or more particles are produced, at least one of which is 
different from the incident particles. Elastic scattering is always possible, 
but reactions may or may not be energetically possible, depending on the 
differences in masses of the particles and the incident energy. To determine 
the energetics involved and to see the processes in their simplest form 
kinematically it is convenient to transform to a coordinate frame K', 
where the projectile and the target have equal and oppositely directed 
momenta. This frame is called the center of momentum system (sometimes, 
loosely, the center of mass system) and is denoted by CM system. The 
scattered particles (or reaction products in a two-body reaction) have 
equal and opposite momenta making an angle 0' with the initial momenta. 



398 



Classical Electrodynamics 



Figure 12.2 shows the momentum vectors involved in elastic scattering or 
a two-body reaction. For elastic scattering, |p'| = |q'|, but for a reaction 
the magnitude of q' must be determined from conservation of total energy 
(including rest energies) in the CM system. 

To relate the incident energy and momentum in the laboratory to the 
CM variables we can either make a direct Lorentz transformation to K', 
determining the transformation velocity v CM from the requirement that 
p/ = p' = —p 2 ', or we can use the invariance of scalar products. Adopting 
the latter procedure, we consider the invariant scalar product 



(Pi + P2) • (Pi + P2) = (Pi + P2) • (Pi + p 2 ') 



(12.29) 



The left-hand side is to be evaluated in the laboratory, where p 2 = 0, and 
the right-hand side in the CM system, where p/ + p 2 ' = 0. Consequently 
we obtain 

f - (E 1 + m 2 ) 2 = -(£/ + E 2 'f (12.30) 

Using E ± 2 = p 2 + rrij 2 , we find that the total energy in the CM system is 

E' = E{ + E 2 ' = (mj 2 + m 2 + 2E x m^ (12.31) 

The separate energies E-l and E 2 ' can be found by considering scalar 
products like 

Pi ' (Pi + P*) = Pi ' (Pi + P2) (12.32) 

This gives 



Similarly 



E 1 ' = 



E' = 



E' 2 + m/ - m 2 2 

IE' 
E' 2 + m 2 2 - m/ 

IE' 



(12.33) 



We note the similarity of these expressions to (12.21) and (12.22). The 
magnitude of the momentum p' can be obtained from (12.33): 



m 2 p 
E' 



(12.34) 




Fig. 12.2 Momentum vectors in the center 

of momentum frame for elastic scattering or 

a two-body reaction. 



[Sect. 12.3] Relativistic-Particle Kinematics and Dynamics 399 

The Lorentz transformation parameters v CM and y CM can be found by 
noting that p 2 ' = -/cm^cm = -p' and E t = rcM"V This g ives 

El + m2 (12.35) 



7cm ~ 



* CM £ 1 + m 2 ' '^ E' 

For nonrelativistic motion the kinetic energy in the CM system reduces 

T = E' - (m, + m 2 ) - ( ™ 2 \\mtf (12.36) 

\m x + m 2 / 

Similarly the CM velocity and the momentum in the CM system are 

v CM = (-=»-k W = (-^W (12.37) 

CM Xmj + mj \m x + m 2 / 

We see that we can recover the familiar nonrelativistic results from our 
completely relativistic expressions. In the extreme of ultrarelativistic 
motion (E x > m 1 and m 2 ) the various quantities take on the approximate 
limiting values: E'-QE^* 



CM_ E, 



(12.38) 



The energy available in the CM system is seen to increase only as the square 
root of the incident energy. This means that it is very difficult to obtain 
ultrahigh energies in the CM frame when bombarding stationary targets. 
The highest-energy accelerators presently existing (at CERN, near Geneva, 
Switzerland, and at Brookhaven, N.Y.) produce protons of approximately 
30 Bev. If the target is a stationary nucleon, this means about 7 Bev total 
CM energy. To have 30 Bev available in the CM frame it would be 
necessary to bombard a stationary nucleon with protons of over 470 Bev ! 
Considerable effort is being put into designs for so-called colliding or 
clashing beam accelerators so that no energy is wasted in CM motion. 

In a reaction the initial particles of mass m x and m 2 are transformed into 
two or more particles with masses /Wj (i = 3,4, . . .). Let AM be the 
difference between the sum of masses finally and the sum of masses 

initially:* AM = (m 3 + m 4 + • • •) - K + in.) (12.39) 

If AM is positive, the reaction will not occur below a certain incident 
kinetic energy T th , called the threshold for the reaction. The criterion for 
the reaction just to occur is that there be enough energy available in the 

* Note that this definition of AM is the negative of the one used in Section 12.2 for 
decay processes. 



400 Classical Electrodynamics 

CM system that the particles can be created with no kinetic energy. This 
means that 

(E'\ h = m 1 + w 2 + A¥ (12.40) 

Using (12.31), it is easy to show that the incident kinetic energy of the 
projectile at threshold is 

T„, = AM (l + Si + ^) (12.41) 



The first two terms in the parentheses are the nonrelativistic terms, while 
the last is a relativistic contribution. To illustrate the reaction-threshold 
formula we consider the calculation of the threshold energy for photo- 
production of neutral pi mesons from protons : 

y + p^p + ^ 

Since the photon has no rest mass, the mass difference is AM == m^ = 
135.0 Mev, while the target mass is m 2 = m p = 938.5 Mev. Then the 
threshold energy is 

135.0 ~ 



T th = 135.0 



1 + 



2(938.5). 



= 135.0(1.072) = 144.7 Mev 



As another example consider the production of a proton-antiproton pair 
in proton-proton collisions : 

p+p^p+p+p+p 

The mass difference is AM-= 2m v = 1.877 Bev. From (12.41) we find 

r th = 2m p (l + 1 + 1) = 6m p = 5.62 Bev 

In this example we find a factor-of-3 increase over the actual mass 
difference, whereas in the photoproduction example the increase was only 
7.2 per cent. Other threshold calculations are left to Problem 12.1. 



12.4 Transformation of Scattering or Reaction Momenta and Energies 
from CM to Laboratory System 

In Fig. 12.2 the various CM momenta for a two-body collision are 
shown. The initial momenta and energies (p/ = — p 2 ' = p', E x \ E 2 ') have 
already been calculated, (12.33) and (12.34). The final CM momenta and 
energies (p 3 ' = -p 4 ' = q', E 3 ', E^) can be calculated similarly. Since 
energy and momentum are conserved, the 4-vector momenta satisfy 

Pi +P* =Pz +P* (12.42) 



[Sect. 12.4] Relativistic-Particle Kinematics and Dynamics 401 

Then it is easy to show that the energies of the outgoing particles are 



£3' = 
£4' = 



E* + m 3 2 - ml 

IE' 
E' 2 + m 4 2 - m? 

IE' 



(12.43) 



where E' is given by (12.31). The obvious symmetry with (12.33) should 
be noted. The CM momentum of the outgoing particles is 

"' = f It 1 - (H^J] i 1 - (")T (1144) 



An alternative form of this result is 

AE 



E' 



•L Ex + 2?W)~ 
\ m 9 / - 



(12.45) 



where A£ x is the incident projectile's energy in the laboratory above the 
threshold energy (12.41): 

AE, = T X - T th (12.46) 

For elastic scattering where m 3 = m x , m 4 = m 2 , (12.45) obviously reduces 
to (12.34). 

Since the scattering or reaction is actually observed in the laboratory, it 
is necessary to transform back from the CM frame to the laboratory. 
Figure 12.3 shows the initial momentum p and the final momenta p 3 and p 4 
in the laboratory. The CM momenta in Fig. 12.2 have been thrown 
forward by the Lorentz transformation. We can express the laboratory 
energy E 3 in terms of CM quantities by the Lorentz transformation v CM , 
using (12.35) and (12.4). If 6' is the CM angle of p 3 ' with respect to the 
incident direction, we find 



E z = y C M(£ 3 ' + ^cm?' cos 6') 



(12.47) 



mi«- 



Fig. 12.3 Momentum vectors 

in laboratory for a two-body 

process. 




402 Classical Electrodynamics 

Then an explicit expression is 



£3 = XPi + m 2 )[ 1 + 



m a 



7/ 2 J 



+ 



1 1 1 _ / m 3 + m 4 



1 - 



m, — 



T^jf j cos 0' (12.48) 



where £' is given by (12.31). To obtain E A we merely interchange m 3 and 
m 4 and change 0' into n — 6' (cos 0' -> —cos 0'). 

The relation between angles 0' and 3 can be obtained from the expres- 
sion 

tan0, = ^ = l' sind ' 



Therefore we find 



tan 0„ = 



Pm 7cm(«' cos0 ' + ^cm £ 3) 
E' sin 6' 



where 



a = 



_ vcmEs' 



{E x + m 2 )(cos 0' + a) 
1 + 



(12.49) 
(12.50) 



m 3 2 — m 4 2 



£' 2 



£ i + m 2 f|"i _ / ma + m^ ir _ fa, - m 4 \ 2 "|j' 



(12.51) 
We note that a is the ratio of the CM velocity to the velocity of particle 3 
in the CM system. Just above threshold, a will be large compared to unity. 
This means that, as 6' ranges over all values from -► 77 in the CM system, 
3 will be confined to some forward cone, < 3 < max . Figure 12.4 
shows the general behavior when a > 1 . The laboratory angle 3 is double 
valued if a > 1, with particles emitted forwards and backwards in the CM 
system appearing at the same laboratory angle. The two types of particles 
can be distinguished by their energies. From (12.48) it is evident that the 
particles emitted forwards in the CM frame will be of higher energy than 
those emitted backwards. For a < 1, it is evident that the denominator in 
(12.50) can vanish for some 0' > (tt/2), implying 3 = (tt/2), and is 
negative for large 0'. This means that 3 varies over the full range 
(0 < 3 < 77-) and is a single-valued function of 0'. Such a curve is shown 
in Fig. 12.4. 

Although it is not difficult to relate 0' and 3 through (12.50) and so 
obtain E z as a function of 3 from (12.48), it is sometimes convenient to 
have an explicit expression for this relationship. Using conservation of 
energy and momentum in the laboratory, 



Px+ P2=P*+ Pi 



(12.52) 



[Sect. 12.4] Relativistic-Particle Kinematics and Dynamics 



403 



Fig. 12.4 Laboratory angle 3 

of particle 3 versus center of 

momentum angle 6' for a < 1 

and a > 1. 




it is a straightforward, although tedious, matter to obtain the result: 

( ,t? . 4 it ■ "h 2 + ™2 2 + ™* - m? \ 
(E 1 + mJlmiEi H ^ J 



±p cos o 3 



E % = 



[(«A + =^ 



+ m 2 2 — m 3 2 — m 4 



)'- 



m z 2 m 4 2 — p 2 m 3 * sin 2 o 3 



J* 



(£ x + /n 2 ) 2 - p 2 cos 2 3 



(12.53) 



Only the values of (12.53) greater than m 3 have physical significance. Both 
roots are allowed when a > 1 in (12.50), but only one when a < 1, as can 
be readily verified. To obtain £ 4 we merely interchange m z and m 4 and 
replace 6 3 by 4 . 

For elastic scattering with m 3 = m x , m 4 = m 2 , the above relationships 
simplify considerably. The scattering angle in the laboratory is given by 
(12.50) with 



m 1 \m 1 



+ m 1 



m 2 (E 1 + m 2 ) 



(12.54) 



In the nonrelativistic limit this reduces to the well-known result a = m x \m 2 . 
The energy lost by the incident particle is A£ = T 4 = E± — m 4 . From 
(12.48) we can obtain AE in terms of the CM scattering angle: 



AE = 



m 2 p 2 (l — cos B') 
2m 2 E 1 + m x 2 + m 2 2 



(12.55) 



404 



Classical Electrodynamics 



An alternative expression for A£in terms of the laboratory angle of recoil 
4 can be found from (12.53): 

AE = 2m 2 p 2 cos 2 6 4 

2m 2 E 1 + m* + m 2 2 + f sin 2 4 U °' 

For a head-on collision both expressions take on the maximum value 



A£ max = ^— H2 57) 

The nonrelativistic value of A^^ is 

A£m ^(^w (imi,,l2) (12 - 58) 

showing that all the incident kinetic energy can be transferred in a head-on 
collision if m 1 = m 2 (true relativistically as well). 

An important example of energy transfer occurs in collisions between 
incident charged particles and atomic electrons. These electrons can be 
treated as essentially at rest. If the incident particle is not an electron, 
m x > m 2 . Then the maximum energy transfer can be written approxi- 
mately as 

A£ max ~ 2m 2 \^-J = 2m 2 y 2 p 2 (12.59) 

where y, p are characteristic of the incident particle. Equation (12.59) is 
valid, provided the incident energy is not too large : 



£i < (— )m 1 (12.60) 

\m 2 / 



For mu mesons this limit is 20 Bev ; for protons it is nearly 2000 Bev. For 
electron-electron collisions {m x = m 2 = m), the maximum energy transfer 
is 

A£Sx = (y - l)m (12.61) 



12.5 Covariant Lorentz Force Equation; Lagrangian and Hamiltonian 
for a Relativistic Charged Particle 

In Section 12.1 we considered the Lorentz force equation as a method 
of establishing the Lorentz transformation properties of the momentum 
and energy of a particle, but we did not explicitly examine the equation as 
a covariant equation of motion for a particle moving in external fields. We 



[Sect. 12.5] Relativistic-Particle Kinematics and Dynamics 405 

now want to establish that covariance and discuss the associated Lagran- 
gian, canonical momenta, and Hamiltonian. From equations (12.2) and 
and (11.129) we see that we can write the force equation in the form: 



dt c J 



F UV J V d 3 x (12.62) 



where the volume integral is over the extent of the charge. If the particle's 
velocity is v and its total charge e, then 

^ = -F MV t; v (12.63) 

dt c 

where v v = v k for v = k = 1, 2, 3 and u 4 = ic. This is not yet a co variant 
form for the equation, since v v is not a 4-vector, and dpjdt is not one 
either. This deficiency can be remedied by writing a derivative with 
respect to proper time t (11.62) rather than t. Since dt = y dr, we obtain 

^ = -iWt; v (12.64) 

dr c 

But now yv v = pjm is a 4-vector (sometimes called the 4-velocity). Con- 
sequently we arrive at the obviously co variant force equation for a particle : 

^ = -FW> v (12.65) 

dr mc 

This is the counterpart for a discrete particle of the Lorentz force-density 
equation (11.129) for continuously distributed charge and current. 
Having established its covariance, it is often simplest to revert to the 
space-time forms : 



dt \ c 

dE ,-, 

— = e\ • E 

dt 



(12.66) 



in any convenient reference frame. Equation (12.65) shows that, as long 
as all the different quantities are transformed according to their separate 
transformation laws, the noncovariant forms will be valid in any Lorentz 

frame. 

Although the force equation (12.65) or (12.66) is sufficient to describe 
the general motion of a charged particle in external electromagnetic fields, 
it is sometimes convenient to use the ideas and formalism of Lagrangian 
and Hamiltonian mechanics. In order to see how to obtain an appropriate 
Lagrangian for the Lorentz force equation, we start with a free, but 



406 Classical Electrodynamics 

relativistic, particle. Since the Lagrangian must be a function of velocities 
and coordinates, we write the free-particle equation of motion as 

— (ymy) = (12.67) 

where y = [1 - (v 2 /c 2 )]-^. At the least sophisticated level we know that 
the Lagrangian L must be chosen strategically so that the Euler-Lagrange 
equations of motion, 

d (dL\ dL „ 

ifa)-*,- (12 - 68) 

are the same as Newton's equations of motion. Only a moment's con- 
sideration shows that a suitable Lagrangian for a free particle is 

L f =-mc 2 \l--J (12.69) 

Evidently this form yields (12.67) when substituted into (12.68). 

To obtain the free-particle Lagrangian in a more elegant way we 
consider Hamilton's principle or the principle of least action. This 
principle states that the motion of a mechanical system is such that in 
going from one configuration a at time t x to another configuration b at 
time t 2 , the action integral A, defined as the time integral of the Lagrangian 
along the path of the system, 



-J*' 



A = \ Ldt (12.70) 

J a 

is an extremum (actually a minimum). By considering small variations of 
the path taken and demanding 6A = 0, one obtains the Euler-Lagrange 
equations of motion (12.68). We now appeal to the Lorentz invariance of 
the action in order to determine the free-particle Lagrangian. That the 
action is a Lorentz scalar follows the first postulate of relativity, since the 
requirement that it be an extremum determines the mechanical equations 
of motion. If we introduce the proper time through dt = y dr, the action 
integral becomes : 

(yL) dr (12.71) 

Since proper time is Lorentz invariant, the condition that A be also 
Lorentz invariant forces yL to be Lorentz invariant. This is a general 
condition on the Lagrangian. For a free particle L f can be a function of 
only the velocity of the particle (and perhaps its mass). The only Lorentz 
invariant quantities involving the velocity are functions of the 4-vector 
scalar product (l/m 2 )(p • p), where p^ is the 4-momentum of the particle. 






[Sect. 12.5] Relativistic-Particle Kinematics and Dynamics 407 

Since (p ■ p) = —m 2 , we see that for a free particle yL f is a constant, 

yL f = -I (12.72) 

Then the action is proportional to the integral of the proper time over the 
path from the initial space-time point a to the final space-time point b. 
This integral is Lorentz invariant, but depends on the path taken. For 
purposes of calculation, consider a reference frame in which the particle 
is initially at rest. From definition (11.62) of proper time it is clear that, 
if the particle stays at rest in that frame, the integral over proper time will 
be larger than if it moves with a nonzero velocity along its path. Con- 
sequently we see that a straight world line joining the initial and final 
points of the path gives the maximum integral over proper time, or, with 
the negative sign in (12.72), a minimum for the action integral. Com- 
parison with Newton's equation for nonrelativistic motion shows that 
X = mc 2 , yielding the free-particle Lagrangian (12.69). 

The general requirement that yL be Lorentz invariant allows us to 
determine the Lagrangian for a relativistic charged particle in external 
electromagnetic fields, provided we know something about the Lagrangian 
(or equations of motion) for nonrelativistic motion in static fields. A 
slowly moving charged particle is influenced predominantly by the electric 
field which is derivable from the scalar potential O. The potential energy 
of interaction is V = eO. Since the nonrelativistic Lagrangian is (T — V), 
the interaction part L iQt of the relativistic Lagrangian must reduce in the 
nonrelativistic limit to 

Lmt — L?S = -e<D (12.73) 

Our problem thus becomes that of finding a Lorentz invariant expression 
for yL int which reduces to (12.73) for nonrelativistic velocities. Since O 
is the fourth component of the 4-vector potential A M , we anticipate that 
yL int will involve the scalar product of A^ with some 4-vector. The only 
other 4-vectors available are the momentum and position vectors of the 
particle. Since gamma times the Lagrangian must be translationally 
invariant as well as Lorentz invariant, it cannot involve the coordinates 
explicitly. Hence the interaction Lagrangian must be* 

Lmt = - — (p ■ A) = ^^ - e<D (12.74) 

y mc c 

* Without appealing to the nonrelativistic limit this form of L lnt can be written down 
by demanding that yL lDt be a Lorentz invariant which is (1) linear in the charge of the 
particle, (2) linear in the electromagnetic potentials, (3) translationally invariant, and 
(4) a function of no higher than the first time derivative of the particle coordinates. The 
reader may consider the possiblity of an interaction Lagrangian satisfying these condi- 
tions, but linear in the field strengths F^, rather than the potentials A^. 



408 Classical Electrodynamics 

where the coefficient of the scalar product (p • A) is chosen to yield (12.73) 
in the limit v — ► 0. 

The combination of (12.69) and (12.74) yields the complete relativistic 
Lagrangian for a charged particle : 



ri 



l= \ 



— mc 2 H (p- A) 

yL mc 



(12.75) 



-mc 2 i _l + -vA-eO 



where the upper (lower) line gives L in 4-vector (explicit space-time) form. 
Verification that (12.75) does indeed lead to the Lorentz force equation 
will be left as an exercise for the reader. Use must be made of the con- 
vective derivative [d/dt = (d/dt) + v • V] and the standard definitions of 
the fields in terms of the potentials. 

The canonical momentum P conjugate to the position coordinate x is 
obtained by the definition, 

P i = — = ymv i + -A i (12.76) 

dv i c 

Thus the conjugate momentum is 

P = p + -A (12.77) 

c 

where p = ym\ is the momentum in the absence of fields. The Hamiltonian 
H is a function of the coordinate x and its conjugate momentum P and is a 
constant of the motion if the Lagrangian is not an explicit function of 
time. The Hamiltonian is defined in terms of the Lagrangian as 

H = P • v - L (12.78) 

The velocity v must be eliminated from (12.78) in favor of P and x. From 
(12.76) or (12.77) we find that 

cP - eA 



v = 



-f) 



^ 22 (12-79) 



When this is substituted into (12.78) and into L (12.75), the Hamiltonian 
takes on the form : 

H = V(cP - eA) 2 + w 2 c 4 + e<S> (12.80) 

Again the reader may verify that Hamilton's equations of motion can be 
combined to yield the Lorentz force equation. Equation (12.80) is an 



[Sect. 12.6] Relativistic-Particle Kinematics and Dynamics 409 

expression for the total energy W of the particle. It differs from the free- 
particle energy by the addition of the potential energy e® and by the 
replacement p -> [P — 0/c)A]. These two modifications are actually only 
one 4- vector change. This can be seen by transposing eO in (12.80) and 
squaring both sides. Then 

(cP - eXf -{W- e<S>f = -{mc*f (12.81) 

This is just the 4-vector scalar product 

(p.p)=-(mcf (12.82) 



where ft _(^).[(,_2).v_ 



eO) 



(12.83) 



We see that in some sense the total energy Wis the fourth component of a 
canonically conjugate 4-momentum of which (12.77) is the space part. An 
alternative formulation with a relativistically invariant Lagrangian which 
is a function of the 4- velocity u M = pjm is discussed in Problem 12.5. There 
the canonical 4-momentum arises naturally. 

The Lagrangian and Hamiltonian formulation of the dynamics of a 
charged particle has been outlined for several reasons. One is that the 
concept of Lorentz invariance, coupled with other physical requirements, 
was shown to be a powerful tool in the systematic construction of a 
Lagrangian which yields dynamic equations of motion. Another is that 
the Lagrangian is often a convenient starting point in discussing particle 
dynamics. Finally, the concepts and ideas of conjugate variables, etc., 
are useful even when one proceeds to solve the force equation directly. 



12.6 Lowest-Order Relativistic Corrections to the Lagrangian for 
Interacting Charged Particles 

In the previous section we discussed the general Lagrangian formalism 
for a relativistic particle in external electromagnetic fields described by the 
vector and scalar potentials, A and O. The appropriate interaction 
Lagrangian was given by (12.74). If we now consider the problem of a 
Lagrangian description of the interaction of two or more charged particles 
with each other, we see that it is possible only at nonrelativistic velocities. 
The Lagrangian is supposed to be a function of the instantaneous velocities 
and coordinates of all the particles. When the finite velocity of propa- 
gation of electromagnetic fields is taken into account, this is no longer 
possible, since the values of the potentials at one particle due to the other 
particles depend on their state of motion at "retarded" times. Only when 



•410 Classical Electrodynamics 

retardation effects can be neglected is a Lagrangian description of the 
system of particles alone possible. In view of this one might think that a 
Lagrangian could be formulated only in the static limit, i.e., to zeroth 
order in (v/c). We will now show, however, that lowest-order relativistic 
corrections can be included, giving an approximate Lagrangian for inter- 
acting particles, correct to the order of (v/c) 2 inclusive. 

It is sufficient to consider two interacting particles with charges q x and 
q 2 , masses m 1 and m 2 , and coordinates x x and x 2 . The relative separation 
is r = x x — x 2 . The interaction Lagrangian in the static limit is just the 
negative of the electrostatic potential energy, 

L NR = _ M2 (12g4) 

r 

If attention is directed to the first particle, this can be viewed as the negative 
of the product of q x and the scalar potential 12 due to the second particle 
at the position of the first. This is of the same form as (12.73). If we wish 
to generalize beyond the static limit, we must, according to (12.74), 
determine both 12 and A 12 , at least to some degree of approximation. In 
general there will be relativistic corrections to both <I> 12 and A 12 . But in 
the Coulomb gauge, the scalar potential is given correctly to all orders in 
v/c by the instantaneous Coulomb potential. Thus, if we calculate in that 
gauge, the scalar-potential contribution 12 is already known. All that 
needs to be considered is the vector potential A 12 . 

If only the lowest-order relativistic corrections are desired, retardation 
effects can be neglected in computing A 12 . The reason is that the vector 
potential enters the Lagrangian (12.74) in the combination q x (y x lc) • A 12 . 
Since A 12 itself is of the order of vjc, greater accuracy in calculating A 12 
is unnecessary. Consequently, we have the magnetostatic expression, 

^luwdv (1285) 

c J \x x — x'l 

where J t is the transverse part of the current due to the second particle, 
as discussed in Section 6.5. From equations (6.46)-(6.50) it is easy to see 
that the transverse current is 

J t (x') = 4 2 v 2 *x' - x 2 ) - q f V ( V2 : (X '~f ) (12.86) 

Att \ |x — X«5| / 

When this is inserted in (12.85), the first term can be integrated immediately. 
Thus 

A ~ 42y_ 2 _ qz_ f 1 v , / v 2 -(x'-x 2 ) \ dZx , 

12 cr Attc J |x' - Xl | \ |x' - x 2 | 3 / 



[Sect. 12.7] Relativistic-Particle Kinematics and Dynamics 411 

By changing variables to y = x' - x 2 and integrating by parts, the integral 
can be put in the form, 

A qft _ 31. V r f ^ -i— d*y (12.88) 

cr 4ttc J f |y - r| 

The integral can now be done in a straightforward manner to yield 



A ~?2 

■i\.ia 



2 v r 

Lr 



(¥)1 



The differentiation of the second term leads to the final result 

r(y 2 • r) 



g2 

2cr 



v 2 + 



(12.89) 



(12.90) 



With expression (12.90) for the vector potential due to the second 
particle at the position of the first, the interaction Lagrangian for two 
charged particles, including lowest-order relativistic effects, is 



r _ 4i<7 2 ( 1 , J_ 



v . v , (vi-r)(v 2 .ry 
1 r 2 



(12.91) 



This interaction form was first obtained by Darwin in 1920. It is of 
importance in a quantum-mechanical discussion of relativistic corrections 
in two-electron atoms. In the quantum-mechanical problem the velocity 
vectors are replaced by their corresponding quantum-mechanical operators 
(Dirac a's). Then the interaction is known as the Breit interaction (1930). 



12.7 Motion in a Uniform, Static, Magnetic Field 

As a first important example of the dynamics of charged particles in 
electromagnetic fields we consider the motion in a uniform, static, 
magnetic induction B. The equations of motion (12.66) are 



dp e _ dE _ 
K = -vxB, — = 



dt 



dt 



(12.92) 



Since the energy is constant in time, the magnitude of the velocity is 
constant and so is y. Then the first equation can be written 



d\ 
dt 



= V X CO] 



where 



t»B = 



eB _ ecB 

ymc E 



(12.93) 
(12.94) 



412 Classical Electrodynamics 

is the gyration or precession frequency. The motion described by (12.93) 
is a circular motion perpendicular to B and a uniform translation parallel 
to B. The solution for the velocity is easily shown to be 

v(t) = v u e 3 + co B a(e 1 - it^e'™* 1 (12.95) 

where c 3 is a unit vector parallel to the field, e x and e 2 are the other 
orthogonal unit vectors, y„ is the velocity component along the field, and 
a is the gyration radius. The convention is that the real part of the equation 
is to be taken. Then one can see that (12.95) represents a counterclockwise 
rotation (for positive charge e) when viewed in the direction of B. Another 
integration yields the displacement of the particle, 

x(r) = Xo + v H te 3 + iafa - i^)e- iu>Bt (12.96) 

The path is a helix of radius a and pitch angle a = tan -1 (vJa> B a). The 
magnitude of the gyration radius a depends on the magnetic induction B 
and the transverse momentum p x of the particle. From (12.94) and (12.95) 
it is evident that 

cp L = eBa 

This form is convenient for the determination of particle momenta. The 
radius of curvature of the path of a charged particle in a known B allows 
the determination of its momentum. For particles with charge the same 
in magnitude as the electronic charge, the momentum can be written 
numerically as 

p ± (Mev/c) = 3.00 x lO" 4 ^ (gauss-cm) (12.97) 



12.8 Motion in Combined, Uniform, Static Electric and Magnetic Fields 

We now consider a charged particle moving in a combination of electric 
and magnetic fields E and B, both uniform and static, but in general not 
parallel. As an important special case, perpendicular fields will be treated 
first. The force equation (12.66) shows that the particle's energy is not 
constant in time. Consequently we cannot obtain a simple equation for 
the velocity, as was done for a static magnetic field. But an appropriate 
Lorentz transformation simplifies the equations of motion. Consider a 
Lorentz transformation to a coordinate frame K' moving with a velocity 
u with respect to the original frame. Then the Lorentz force equation for 
the particle in K' is 

d^ 
dt' 



t' \ C 1 



[Sect. 12.8] Relativistic-Particle Kinematics and Dynamics 413 

where the primed variables are referred to the system K'. The fields E' 
and B' are given by relations (11.115) with v replaced by u, where || and i 
refer to the direction of u. Let us first suppose that |E| < |B|. If u is 
now chosen perpendicular to the orthogonal vectors E and B, 



u = c 



(E x B) 
B 2 



(12.98) 



we find the fields in K' to be 



(12.99) 



E„' = 0, E x ' = ylE + - xBl =0 

b,'-o, b.'-Ib-P^J'b 

In the frame K' the only field acting is a static magnetic field B' which 
points in the same direction as B, but is weaker than B by a factor y~ x . 
Thus the motion in K' is the same as that considered in the previous 
section, namely a spiraling around the lines of force. As viewed from the 
original coordinate system, this gyration is accompanied by a uniform 
"drift" u perpendicular to E and B given by (12.98). This drift is sometimes 
called the E x B drift. It has already been considered for a conducting 
fluid in another context in Section 10.3. The drift can be understood 
qualitatively by noting that a particle which starts gyrating around B is 
accelerated by the electric field, gains energy, and so moves in a path with 
a larger radius for roughly half of its cycle. On the other half, the electric 
field decelerates it, causing it to lose energy and so move in a tighter arc. 
The combination of arcs produces a translation perpendicular to E and B 
as shown in Fig. 12.5. The direction of drift is independent of the sign of 
the charge of the particle. 

The drift velocity u (12.98) has physical meaning only if it is less than 
the velocity of light, i.e., only if |E| < |B|. If |E| > |B|, the electric field 






Fig. 12.5 E x B drift of charged 
particles in crossed fields. 



414 Classical Electrodynamics 

is so strong that the particle is continually accelerated in the direction of E 
and its average energy continues to increase with time. To see this we 
consider a Lorentz transformation from the original frame to a system K" 
moving with a velocity 



(Ex B) 
£ 2 



u' = c ^-^ (12.100) 



relative to the first. In this frame the electric and magnetic fields are 

1 /f 2 — R 2 Y^ 

v = o, e/ = -e=(^)e 



B„" = 0, B x " = y'\B - Hl^Zj = o 



(12.101) 



Thus in the system K" the particle is acted on by a purely electrostatic 
field which causes hyperbolic motion with ever-increasing velocity (see 
Problem 12.7). 

The fact that a particle can move through crossed E and B fields with 
the uniform velocity u = cE/B provides the possibility of selecting charged 
particles according to velocity. If a beam of particles having a spread in 
velocities is normally incident on a region containing uniform crossed 
electric and magnetic fields, only those particles with velocities equal to 
cEjB will travel without deflection. Suitable entrance and exit slits will 
then allow only a very narrow band of velocities around cE/B to be 
transmitted, the resolution depending on the geometry, the velocities 
desired, and the field strengths. When combined with momentum 
selectors, such as a deflecting magnet, these E x B velocity selectors can 
separate a very pure and monoenergetic beam of particles of a definite 
mass from a mixed beam of particles with different masses and momenta. 
Large-scale devices of this sort are commonly used to provide experimental 
beams of particles produced in very high-energy accelerators. 

If E has a component parallel to B, the behavior of the particle cannot 
be understood in such simple terms as above. The scalar product E • B is a 
Lorentz invariant quantity (see Problem 1 1.10), as is (2? 2 — E 2 ). When the 
fields were perpendicular (E • B = 0), it was possible to find a Lorentz 
frame where E = Oif |B| > |E|,orB = Oif |E| > |B|. In those coordinate 
frames the motion was relatively simple. If E -B^0, electric and magnetic 
fields will exist simultaneously in all Lorentz frames, the angle between the 
fields remaining acute or obtuse depending on its value in the original 
coordinate frame. Consequently motion in combined fields must be 
considered. When the fields are static and uniform, it is a straightforward 
matter to obtain a solution for the motion in cartesian components. This 
will be left for Problem 12.10. 



[Sect. 12.9] Relativistic-Particle Kinematics and Dynamics 



415 



12.9 Particle Drifts in Nonuniform, Static Magnetic Fields 



In astrophysical and thermonuclear applications it is of considerable 
interest to know how particles behave in magnetic fields which vary in 
space. Often the variations are gentle enough that a perturbation solution 
to the motion, first given by Alfven, is an adequate approximation. 
"Gentle enough" generally means that the distance over which B changes 
appreciably in magnitude or direction is large compared to the gyration 
radius a of the particle. Then the lowest-order approximation to the 
motion is a spiraling around the lines of force at a frequency given by the 
local value of the magnetic induction. In the next approximation, slow 
changes occur in the orbit which can be described as a drifting of the 
guiding center. 

The first type of spatial variation of the field to be considered is a 
gradient perpendicular to the direction of B. Let the gradient at the point 
of interest be in the direction of the unit vector n, with n • B = 0. Then, 
to first order, the gyration frequency can be written 



Wb(x) = 



ymc 



B(x) 



tOr 



1 + 



Bo Wo 



(12.102) 



In (12.102) £ is the coordinate in the direction n, and the expansion is 
about the origin of coordinates where a> B = co . Since the direction of Bis 
unchanged, the motion parallel to B remains a uniform translation. 
Consequently we consider only modifications in the transverse motion. 
Writing v ± = v + v 1? where v is the uniform-field transverse velocity and 
v x is a small correction term, we can substitute (12.102) into the force 
equation 



— ± = Vj. X U) B (X) 

dt 



(12.103) 



and, keeping only first-order terms, obtain the approximate result, 



d\ 1 
dt 



_l / ^ ( dB \ 
v 1 + v (n.x )— — I 

B n \ dt /, 



B \df /„. 



X tO n 



(12.104) 



From (12.95) and (12.96) it is easy to see that for a uniform field the 
transverse velocity v and coordinate x are related by 



v = -w x (x - X) 



(Xq - X) = — (co x v ) 



(12.105) 



416 Classical Electrodynamics 

where X is the center of gyration of the unperturbed circular motion 
(X = here). If (o> x v ) is eliminated in (12.104) in favor of x , we 
obtain 

dt 



1 ~ y 1 - — I — I c*) x x (n • x ) x u> (12.106) 

\ L B \d£/ -1 



This shows that, apart from oscillatory terms, \ x has a non zero average 
value, 

y G = <v x > = i- (?f ) to x ((XoUn . x )> (12.107) 

To determine the average value of (x )j(n • x ), it is necessary only to 
observe that the rectangular components of (x ) ± oscillate sinusoidally 
with peak amplitude a and a phase difference of 90°. Hence only the 
component of (x ) ± parallel to n contributes to the average, and 



<(x )i(n • x )> = - n 
Thus the gradient drift velocity is given by 

a 2 1 dB , 



(12.108) 



(12.109) 



An alternative form, independent of coordinates, is 

(12.110) 
From (12.110) it is evident that, if the gradient of the field is such that a 



oi B a 2d 



+e 






VB 



Fig. 12.6 Drift of charged par- 
ticles due to transverse gradient 
of magnetic field. 



[Sect. 12.9] Relativistic-Particle Kinematics and Dynamics 

y 



417 





y 

Bo 












* " ^ 


n£KKK ( <CKKKK < 


bUuu( 


JUUUU x 
















(a) 



(b) 



Fig. 12.7 (a) Particle moving in helical path along lines of uniform, constant magnetic 
induction, (b) Curvature of lines of magnetic induction will cause drift perpendicular 

to the (x, y) plane. 



\VB/B\ < 1, the drift velocity is small compared to the orbital velocity 
(w B a). The particle spirals rapidly while its center of rotation moves slowly 
perpendicular to both B and VB. The sense of the drift for positive 
particles is given by (12.110). For negatively charged particles the sign of 
the drift velocity is opposite ; the sign change comes from the definition 
of co B . The gradient drift can be understood qualitatively from considera- 
tion of the variation of gyration radius as the particle moves in and out of 
regions of larger than average and smaller than average field strength. 
Figure 12.6 shows this qualitative behavior for both signs of charge. 

Another type of field variation which causes a drifting of the particle's 
guiding center is curvature of the lines of force. Consider the two- 
dimensional field shown in Fig. 12.7. It is locally independent of z. On 
the left-hand side of the figure is a constant, uniform magnetic induction 
B , parallel to the x axis. A particle spirals around the field lines with a 
gyration radius a and a velocity co B a, while moving with a uniform velocity 
v u along the lines of force. We wish to treat that motion as a zero-order 
approximation to the motion of the particle in the field shown on the right- 
hand side of the figure, where the lines of force are curved with a local 
radius of curvature R which is large compared to a. 

The first-order motion can be understood as follows. The particle tends 
to spiral around a field line, but the field line curves off to the side. As far 
as the motion of the guiding center is concerned, this is equivalent to a 
centrifugal acceleration of magnitude v^jR. This acceleration can be 
viewed as arising from an effective electric field, 



_ ym R 2 
e R 2 



(12.111) 



418 Classical Electrodynamics 

in addition to the magnetic induction B . From (12.98) we see that the 
combined effective electric field and the magnetic induction cause a 
curvature drift velocity, 

With the definition of co B = eB^ymc, the curvature drift can be written 

y r = llL-(*lL*!>) (12.113) 

co B R\ RB J 

The direction of drift is specified by the vector product, in which R is the 
radius vector from the effective center of curvature to the position of the 
charge. The sign in (12.113) is appropriate for positive charges and is 
independent of the sign of y„. For negative particles the opposite sign 
arises from co B . 

A more straightforward, although pedestrian, derivation of (12.113) can 
be given by solving the Lorentz force equation directly. If we use cylin- 
drical coordinates (p, cf>, z) appropriate to Fig. 1 2.1b with origin at the center 
of curvature, the magnetic induction has only a <f> component, B^ = B . 
Then the force equation can be easily shown to give the three equations, 

p — p(f> 2 = — co B z 

p$ + 2pcf> = (12.114) 

z = a> B p 

If the zero-order trajectory is a helix with radius a small compared to the 
radius of curvature R, then, to lowest order, (j> ~ vJR, while p ~ R. Thus 
the first equation of (12.114) yields an approximate result for z: 

• -■ V " (12.115) 



(OnR 



This is just the curvature drift given by (12.113). 

For regions of space in which there are no currents the gradient drift 
y (12.110) and the curvature drift v c (12.113) can be combined into one 
simple form. This follows from the fact that V x B = implies 

?±2 = - - (12.116) 

BR 2 

Evidently then the sum of v G and \ c is a general drift velocity, 

Vz, = - J ^(^, 2 + K 2 )( R ^) (12.117) 

(o n R \ RB I 



[Sect. 12.10] Relativistic-P article Kinematics and Dynamics 419 

where y = co B a is the transverse velocity of gyration. For singly charged 
nonrelativistic particles in thermal equilibrium, the magnitude of the drift 
velocity is 

^(cm/sec) = p , 17 ^ ( ° K) , (12-118) 

R(m) B(gauss) 

The particle drifts implied by (12.117) are troublesome in certain types 
of thermonuclear machines designed to contain hot plasma. A possible 
configuration is a toroidal tube with a strong axial field supplied by 
solenoidal windings around the torus. With typical parameters of R = 1 
meter, B = 10 3 gauss, particles in a 1-ev plasma (T~ lO^K) will have 
drift velocities v D ^ > 1.8 x 10 3 cm/sec. This means that they will drift out 
to the walls in a small fraction of a second. For hotter plasmas the drift 
rate is correspondingly greater. One way to prevent this first-order drift 
in toroidal geometries is to twist the torus into a figure eight. Since the 
particles generally make many circuits around the closed path before 
drifting across the tube, they feel no net curvature or gradient of the field. 
Consequently they experience no net drift, at least to first order in \\R. 
This method of eliminating drifts due to spatial variations of the magnetic 
field is used in the Stellarator type of thermonuclear machine, in which 
containment is attempted with a strong, externally produced, axial 
magnetic field, rather than a pinch (see Sections 10.5-10.7). 



12.10 Adiabatic Invariance of Flux through Orbit of Particle 

The various motions discussed in the previous sections have been 
perpendicular to the lines of magnetic force. These motions, caused by 
electric fields or by the gradient or curvature of the magnetic field, arise 
because of. the peculiarities of the magnetic-force term in the Lorentz force 
equation. To complete our general survey of particle motion in magnetic 
fields we must consider motion parallel to the lines of force. It turns out 
that for slowly varying fields a powerful tool is the concept of adiabatic 
invariants. In celestial mechanics and in the old quantum theory adiabatic 
invariants were useful in discussing perturbations on the one hand, and in 
deciding what quantities were to be quantized on the other. Our discussion 
will resemble most closely the celestial mechanical problem, since we are 
interested in the behavior of a charged particle in slowly varying fields 
which can be viewed as small departures from the simple, uniform, static 
field considered in Section 12.7. 

The concept of adiabatic invariance is introduced by considering the 
action integrals of a mechanical system. If q t and p+ are the generalized 



420 Classical Electrodynamics 

canonical coordinates and momenta, then, for each coordinate which is 
periodic, the action integral / t is defined by 



Ji^fPidq, (12.119) 

The integration is over a complete cycle of the coordinate q t . For a given 
mechanical system with specified initial conditions the action integrals J { 
are constants. If now the properties of the system are changed in some way 
(e.g., a change in spring constant or mass of some particle), the question 
arises as to how the action integrals change. It can be proved* that, if 
the change in property is slow compared to the relevant periods of motion 
and is not related to the periods (such a change is called an adiabatic 
change), the action integrals are invariant. This means that, if we have a 
certain mechanical system in some state of motion and we make an 
adiabatic change in some property so that after a long time we end up with 
a different mechanical system, the final motion of that different system will 
be such that the action integrals have the same values as in the initial 
system. Clearly this provides a powerful tool in examining the effects of 
small changes in properties. 

For a charged particle in a uniform, static, magnetic induction B the 
transverse motion is periodic. The action integral for this transverse 
motion is 



J = <t>P ± .<H, (12.120) 

where P is the transverse component of the canonical momentum (12.77) 
and d\ is a directed line element along the circular path of the particle. 
From (12.77) we find that 

J = &) ym\ ± • dl + - <b Adl (12.121) 

Since v ± is parallel to dl, we find 



J = d> ymco B a 2 dd + - <P A . d\ (12.122) 

Applying Stokes's theorem to the second integral and integrating over 6 
in the first integral, we obtain 

J = 27rymco B a 2 + - B • n da (12.123) 

c Js 

* See, for example, M. Born, The Mechanics of the Atom, Bell, London (1927). 



[Sect. 12.10] Relativistic-Partick Kinematics and Dynamics 



421 



Since the line element d\ in (12.120) is in a counterclockwise sense relative 
to B, the unit vector n is antiparallel to B. Hence the integral over the 
circular orbit subtracts from the first term. This gives 



J = ymco B 7ra 2 = - (Bira 2 ) 



(12.124) 



making use of a> B = eB/ymc. The quantity B-na 2 is the flux through the 
particle's orbit. 

If the particle moves through regions where the magnetic field strength 
varies slowly in space or time, the adiabatic invariance of /means that the 
flux linked by the particle's orbit remains constant. If B increases, the 
radius a will decrease so that Bna 2 remains unchanged. This constancy of 
flux linked can be phrased in several ways involving the particle's orbit 
radius, its transverse momentum, its magnetic moment. These different 
statements take the forms : 



Ba 2 

yp 



► are adiabatic invariants 



(12.125) 



where \x = {ew B a 2 j2c) is the magnetic moment of the current loop of the 
particle in orbit. If there are only static magnetic fields present, the speed 
of the particle is constant and its total energy does not change. Then the 
magnetic moment /u is itself an adiabatic invariant. In time-varying fields 
or with static electric fields, /u is an adiabatic invariant only in the 
nonrelativistic limit. 

Let us now consider a simple situation in which a static magnetic field 
B acts mainly in the z direction, but has a small positive gradient in that 
direction. Figure 12.8 shows the general behavior of the lines of force. In 
addition to the z component of field there is a small radial component due 
to the curvature of the lines of force. For simplicity we assume cylindrical 
symmetry. Suppose that a particle is spiraling around the z axis in an 




Fig. 12.8 



422 Classical Electrodynamics 

orbit of small radius with a transverse velocity \ xo and a component of 
velocity y ||0 parallel to B at z = 0, where the axial field strength is B . The 
speed of the particle is constant so that at any position along the z axis 

v n 2 + v 1 2 = v ( ? (12.126) 

where v Q 2 = v 10 2 + u ||0 2 is the square of the speed at z = 0. If we assume 
that the flux linked is a constant of the motion, then (12.125) allows us to 
write „ 2 

(12.127) 
o 



B B t 



where B is the axial magnetic induction. Then we find the parallel velocity 
at any position along the z axis given by 

V = V-»Lo 2 ^r (12-128) 

Equation (12.128) for the velocity of the particle in the z direction is 
equivalent to the first integral of Newton's equation of motion for a 
particle in a one-dimensional potential* 

V(z) = \m V -^- B(z) 
B o 
If B(z) increases enough, eventually the right-hand side of (12.128) will 
vanish at some point z = z . This means that the particle spirals in an 
ever-tighter orbit along the lines of force, converting more and more 
translational energy into energy of rotation, until its axial velocity vanishes. 
Then it turns around, still spiraling in the same sense, and moves back in 
the negative z direction. The particle is reflected by the magnetic field, as 
is shown schematically in Fig. 12.9. 

Equation (12.128) is a consequence of the assumption that p 2 \B is 
invariant. To show that at least to first order this invariance follows 
directly from the Lorentz force equation, we consider an explicit solution 
of the equations of motion. If the magnetic induction along the axis is 
B(z), there will be a radial component of the field near the axis given by 
the divergence equation as 

dB(z) 
dz 



B p ( P ,z)~-y u -^ (12.129) 



where p is the radius out from the axis. The z component of the force 
equation is 

(-pj>B.)~-^ ^i>"-=P (12.130) 



ymc Lyme dz 



* Note, however, that our discussion is fully relativistic. The analogy with one- 
dimensional nonrelativistic mechanics is only a formal one. 



[Sect. 12.10] Relativistic-Particle Kinematics and Dynamics 



423 




Fig. 12.9 Reflection of charged 

particle out of region of high field 

strength. 

where <j> is the angular velocity around the z axis. This can be written, 
correct to first order in the small variation of B(z), as 



z~ — 



Vw 2 dB(z) 
2B n dz 



(12.131) 



where we have used p 2 ~ — (a 2 o) B ) = — {v ± qJo) b ^. Equation (12.131) has 
as its first integral equation (12.128), showing that to first order in small 
quantities the constancy of flux Unking the orbit follows directly from the 
equations of motion. 

The adiabatic invariance of the flux linking an orbit is useful in discussing 
particle motions in all types of spatially varying magnetic fields. The 
simple example described above illustrates the principle of the "magnetic 
mirror" : charged particles are reflected by regions of strong magnetic 
field. This mirror property formed the basis of a theory of Fermi for the 
acceleration of cosmic-ray particles to very high energies in interstellar 
space by collisions with moving magnetic clouds. The mirror principle 
can be applied to the containment of a hot plasma for thermonuclear 
energy production. A magnetic bottle can be constructed with an axial 
field produced by solenoidal windings over some region of space, and 
additional coils at each end to provide a much higher field towards the 
ends. The lines of force might appear as shown in Fig. 12.10. Particles 
created or injected into the field in the central region will spiral along the 
axis, but will be reflected by the magnetic mirrors at each end. If the 
ratio of maximum field B m in the mirror to the field B in the central region 
is very large, only particles with a very large component of velocity parallel 
to the axis can penetrate through the ends. From (12.128) is it evident that 
the criterion for trapping is 



M < (*= - if 

v xo I \B / 



(12.132) 



424 



Classical Electrodynamics 




Fig. 12.10 Schematic diagram 
of "mirror" machine for the 
containment of a hot plasma. 



If the particles are injected into the apparatus, it is easy to satisfy require- 
ment (12.132). Then the escape of particles is governed by the rate at 
which they are scattered by residual gas atoms, etc., in such a way that 
their velocity components violate (12.132). 

Another area of application of these principles is to terrestrial and 
stellar magnetic fields. The motion of charged particles in the magnetic 
dipole fields of the sun or earth can be understood in terms of the adiabatic 
invariant discussed here and the drift velocities of Section 12.9. Some 
aspects of this topic are left to Problems 12.11 and 12.12 on the trapped 
particles around the earth (the Van Allen belts). 



REFERENCES AND SUGGESTED READING 

The applications of relativistic kinematics, apart from precision work in low-energy 
nuclear physics, all occur in the field of high-energy physics. In books on that field, the 
relativity is taken for granted, and calculations of kinematics are generally omitted or 
put in appendices. One exception is the book by 

Baldin, Gol'danskii, and Rozenthal 
which covers the subject exhaustively with many graphs. 

The Lagrangian and Hamiltonian formalism for relativistic charged particles is treated 
in every advanced mechanics textbook, as well as in books on electrodynamics. Some 
useful references are 

Corben and Stehle, Chapter 16, 

Goldstein, Chapter 6, 

Landau and Lifshitz, Classical Theory of Fields, Chapters 2 and 3. 
The motion of charged particles in external electromagnetic fields, especially inhomo- 
geneous magnetic fields, is an increasingly important topic in geophysics, solar physics, 
and thermonuclear research. The classic reference for these problems is 

Alfven, 
but the basic results are also presented by 

Chandrasekhar, Chapters 2 and 3, 

Linhart, Chapter 1, 

Spitzer, Chapter 1. 



[Probs. 12] Relativistic-Particle Kinematics and Dynamics 425 

Another important application of relativistic charged-particle dynamics is to high- 
energy accelerators. An introduction to the physics problems of this field will be found 
in 

Corben and Stehle, Chapter 17, 

Livingston. 
For a more complete and technical discussion, with references, consult 

E. D. Courant and H. S. Snyder, Ann. Phys., 3, 1 (1958). 



PROBLEMS 

12.1 Use the transformation to center of momentum coordinates to determine 
the threshold kinetic energies in Mev for the following processes : 

(a) pi-meson production in nucleon-nucleon collisions (m„/M = 0.15), 

(b) pi-meson production in pi meson-nucleon collisions, 

(c) pi-meson pair production in nucleon-nucleon collisions, 
id) nucleon-pair production in electron-electron collisions. 

12.2 If a system of mass M decays or transforms at rest into a number of 
particles, the sum of whose masses is less than M by an amount AM, 

(a) show that the maximum kinetic energy of the /th particle (mass m f ) is 



(7VU,-u«.(i-3-£) 



(b) determine the maximum kinetic energies in Mev and also the ratios 
to AMc 2 of each of the particles in the following decays or transformations 
of particles at rest: 

p -»■ e + v + v 

KT —*■ TfT~ + 1T~ + 77+ 

K± ->e± + 77-° + v 





K± 


-A«± 


+ 7T 


+ v 


p 


+ P 


-*2tt+ 


+ 2tt 


-+7T 


p 


+ P 


— K+ + K~ 


+ 3tt° 



12.3 A pi meson (m^ 2 = 140 Mev) collides with a proton (m 2 c 2 = 938 Mev) 
at rest to create a K meson (m 3 c 2 = 494 Mev) and a lambda hyperon 
(m t c 2 = 1115 Mev). Use conservation of energy and momentum, plus 
relativistic kinematics, to find 

(a) the kinetic energy in Mev of the incident pi meson at threshold for 
production of K mesons, and compare this with the Q value of the 
reaction; 

(b) the kinetic energy of the pi meson in Mev in order to create K 
mesons at 90° in the laboratory; 

(c) the kinetic energy of K mesons emerging at 0° in the laboratory 
when the kinetic energy of the pi meson is 20 per cent greater than in (b); 

id) the kinetic energy of K mesons at 90° in the laboratory when the 
incident pi meson has a kinetic energy of 1500 Mev. 

12.4 It is a well-established fact that Newton's equation of motion 

ma' = eE' 



426 Classical Electrodynamics 

holds for a small charged body of mass m and charge e in a coordinate 
system K' where the body is momentarily at rest. Show that the Lorentz 
force equation 



dt \ c / 



follows directly from the Lorentz transformation properties of accelera- 
tions and electromagnetic fields. 
12.5 An alternative approach to the Lagrangian formalism for a relativistic 
charged particle is to treat the 4-vector of position x M and the 4-velocity 
UfZ = (yv, iyc) as Lagrangian coordinates. Then the Euler-Lagrange 
equations have the obviously covariant form, 



±i d ±\- d ±.=0 



where L is a Lorentz invariant Lagrangian and t is the proper time. 

(a) Show that 

1 q , 

L = - mu^Up + - u^Ap 

gives the correct relativistic equations of motion for a particle interacting 
with an external field described by the 4-vector potential A^. 

(b) Define the canonical momenta and write out the Hamiltonian in 
both covariant and space-time form. The Hamiltonian is a Lorentz 
invariant. What is its value ? 

12.6 (a) Show from Hamilton's principle that Lagrangians which differ only 
by a total time derivative of some function of the coordinates and time are 
equivalent in the sense that they yield the same Euler-Lagrange equations 
of motion. 

(b) Show explicitly that the gauge transformation A^^-Ap + (dKjdx^ 
of the potentials in the charged-particle Lagrangian (12.75) merely 
generates another equivalent Lagrangian. 

12.7 A particle with mass m and charge e moves in a uniform, static, electric 
field E . 

(a) Solve for the velocity and position of the particle as explicit functions 
of time, assuming that the initial velocity v was perpendicular to the 
electric field. 

(Jb) Eliminate the time to obtain the trajectory of the particle in space. 
Discuss the shape of the path for short and long times (define "short" 
and "long" times). 

12.8 It is desired to make an E x B velocity selector with uniform, static, 
crossed, electric and magnetic fields over a length L. If the entrance and 
exit slit widths are Ax, discuss the interval Ah of velocities around the 
mean value u = cEjB, which is transmitted by the device as a function of 
the mass, the momentum or energy of the incident particles, the field 
strengths, the length of the selector, and any other relevant variables. 
Neglect fringing effects at the ends. Base your discussion on the practical 
facts that L ~ few meters, ^x ~3xl0 4 volts/cm, Ax ~ 10^i to 10^ 2 cm, 
u ~ 0.5 to 0.995c. 



[Probs. 12] Relativistic-P article Kinematics and Dynamics 427 

12.9 A particle of mass m and charge e moves in the laboratory in crossed, 
static, uniform, electric and magnetic fields. E is parallel to the x axis; 
B is parallel to the y axis. 

(a) For |E| < |B| make the necessary Lorentz transformation described 
in Section 12.8 to obtain explicitly parametric equations for the particle's 
trajectory. 

(b) Repeat the calculation of (a) for |E| > |B|. 

12.10 Static, uniform electric and magnetic fields, E and B, make an angle of 6 
with respect to each other. 

(a) By a suitable choice of axes, solve the force equation for the motion 
of a particle of charge e and mass m in rectangular coordinates. 

(b) For E and B parallel, show that with appropriate constants of 
integration, etc., the parametric solution can be written 

R 



x = AR sin <j>, y = AR cos <$>, z = - Vl + A 2 cosh (pfi) 

P 

TO 

ct = - Vl + A 2 sinh (p0) 
P 

where R = (mc^leB), p = (E/B), A is an arbitrary constant, and <f> is the 
parameter [actually c/R times the proper time]. 
12.11 The magnetic field of the earth can be represented approximately by a 
magnetic dipole of magnetic moment M = 8.1 x 10 25 gauss-cm 3 . Con- 
sider the motion of energetic electrons in the neighborhood of the earth 
under the action of this dipole field (Van Allen electron belts). 

{a) Show that the equation for a line of magnetic force is r = r sin 2 6, 
where is the usual polar angle (colatitude) measured from the axis of the 
dipole, and find an expression to the magnitude of B along any line of 
force as a function of 0. 

(b) A positively charged particle spirals around a line of force in the 
equatorial plane with a gyration radius a and a mean radius R (a < R). 
Show that the particle's azimuthal position (longitude) changes approxi- 
mately linearly in time according to 



m 



<f>(0=<f>o+^[jjco B (t-t ) 

where (o B is the frequency of gyration at radius R. 

(c) If, in addition to its circular motion of (b), the particle has a small 
component of velocity parallel to the lines of force, show that it undergoes 
small oscillations in around = tt/2 with a frequency Q = (3/ V2)(a/R)(o B . 
Find the change in longitude per cycle of oscillation in latitude. 

(d) For an electron of 10 Mev at a mean radius R = 3 x 10 9 cm, find 
co B and a, and so determine how long it takes to drift once around the 
earth and how long it takes to execute one cycle of oscillation in latitude. 
Calculate these same quantities for an electron of 10 Kev at the same 
radius. 

12.12 A charged particle finds itself instantaneously in the equatorial plane of 
the earth's magnetic field (assumed to be a dipole field) at a distance R 
from the center of the earth. Its velocity vector at that instant makes an 
angle a with the equatorial plane (y u /v x = tan a). Assuming that the 



428 Classical Electrodynamics 

particle spirals along the lines of force with a gyration radius a < i?, and 
that the flux linked by the orbit is a constant of the motion, find an 
equation for the maximum magnetic latitude A reached by the particle as 
a function of the angle a. Plot a graph {not a sketch) of A versus a. Mark 
parametrically along the curve the values of a for which a particle at 
radius R in the equatorial plane will hit the earth (radius R ) for 
R/R = 1.5, 2.0, 2.5, 3, 4, 6, 8, 10. 



13 



Collisions between 
Charged Particles, 
Energy Loss, 
and Scattering 



In this chapter collisions between swiftly moving, charged particles 
are considered, with special emphasis on the exchange of energy between 
collision partners and on the accompanying deflections from the incident 
direction. A fast charged particle incident on matter makes collisions with 
the atomic electrons and nuclei. If the particle is heavier than an electron 
(mu or pi meson, K meson, proton, etc.), the collisions with electrons and 
with nuclei have different consequences. The light electrons can take up 
appreciable amounts of energy from the incident particle without causing 
significant deflections, whereas the massive nuclei absorb very little energy 
but because of their greater charge cause scattering of the incident particle. 
Thus loss of energy by the incident particle occurs almost entirely in 
collisions with electrons. The deflection of the particle from its incident 
direction results, on the other hand, from essentially elastic collisions with 
the atomic nuclei. The scattering is confined to rather small angles, so that 
a heavy particle keeps a more or less straight-line path while losing energy 
until it nears the end of its range. For incident electrons both energy loss 
and scattering occur in collisions with the atomic electrons. Consequently 
the path is much less straight. After a short distance, electrons tend to 
diffuse into the material, rather than go in a rectilinear path. 

The subject of energy loss and scattering is an important one and is 
discussed in several books* where numerical tables and graphs are 

* See references at the end of the chapter. 

429 



430 Classical Electrodynamics 

presented. Consequently our discussion will emphasize the physical ideas 
involved, rather than the exact numerical formulas. Indeed, a full 
quantum-mechanical treatment is needed to obtain exact results, even 
though all the essential features are classical or semiclassical in origin. 
The order of magnitude of the quantum effects are all easily derivable 
from the uncertainty principle, as will be seen in what follows. 

We will begin by considering the simple problem of energy transfer to a 
free electron by a fast heavy particle. Then the effects of a binding force on 
the electron are explored, and the classical Bohr formula for energy loss is 
obtained. Quantum modifications and the effect of the polarization of 
the medium are described, followed by a discussion of energy loss in an 
electronic plasma. Then the elastic scattering of incident particles by 
nuclei and multiple scattering are presented. Finally, a discussion is given 
of the electrical resistivity of a plasma caused by screened Coulomb 
collisions. 



13.1 Energy Transfer in a Coulomb Collision 

A swift particle of charge ze and mass M collides with an electron in an 
atom. If the particle moves rapidly compared to the characteristic velocity 
of the electron in its orbit, during the collision the electron can be treated 
as free and initially at rest. As further approximations we will assume that 
the momentum transfer Ap is sufficiently small that the incident particle 
is essentially undeflected from its straight-line path, and that the recoiling 
electron does not move appreciably during the collision. Then to find the 
energy transfer during the collision we need only calculate the momentum 
impulse caused by the electric field of the incident particle at the position 
of the electron. The particle's magnetic field is of negligible importance if 
the electron is essentially at rest. 

Figure 13.1 shows the geometry of the collision. The incident particle 
has a velocity v and an energy E = yMc 2 . It passes the electron of charge e 
and mass m < M at an impact parameter b. At the position of the elec- 
tron the fields of the incident particle are given by (11.118) with q = ze. 
Only the transverse electric field E x has a nonvanishing time integral. 




[Sect. 13.1] Collisions between Charged Particles 431 

Consequently the momentum impulse A/? is in the transverse direction and 
has the magnitude 

Ap=\ eE x {t)dt = ^- (13.1) 

J- oo bv 

It should be noted that Ap is independent of y, as discussed in Section 
11.10 below Eq. (11.119). The energy transferred to the electron is 

Am =m^(i\ (132) 

2m mv 2 \b 2 / 

The angular deflection of the incident particle is given by ~ A/?//?, 
provided A/? < p. Thus, for small deflections, 

~ — (13.3) 

pub 

This result can be compared with the well-known exact expression for the 
Rutherford scattering of a nonrelativistic particle of charge ze by a 
Coulomb force field of charge z'e: 

2 tan - = (13.4) 

2 pu& 

We see that for small angles the two expressions agree.* 

The energy transfer &E(b) given by (13.2) has several interesting features. 
It depends only on the charge and velocity of the incident particle, not on 
its mass. It varies inversely as the square of the impact parameter so that 
close collisions involve very large energy transfers. There is, of course, an 
upper limit on the energy transfer, corresponding to a head-on collision. 
Our method of calculation is really valid only for large values of b. We 
can obtain a lower limit b min on the impact parameter for which our 
approximate calculation is valid by equating (13.2) to the maximum 
allowable energy transfer (12.59): 

AE(b min ) = A£ max = 2myV (13.5) 

This yields the lower bound, 

bmiQ = J?- (13.6) 

ymv 

* Actually there is a question of reference frames in comparing (13.3) and (13.4). 
Since (13.4) holds for a fixed center of force (or the CM system), we should compare it 
with the result for the deflection of the light electron in the frame where the heavy 
incident particle is at rest. Then (13.3) holds with/? ~ ymv as the electron momentum 
in that frame. The reader may verify that (13.3) and (13.4) are also consistent in the 
frame in which the electron is at rest by using (12.50) and (12.54) to transform angles 
from the CM system to the laboratory. 



432 Classical Electrodynamics 

below which our approximate result (13.2) must be replaced by a more 
exact expression which tends to (13.5) as b -> 0. It can be shown (Problem 
13.1) that a proper treatment gives the more accurate result, 

A£(fr)~^( * ) (13.7) 

mv 2 \fc min + bV 

Equation (13.7) exhibits the proper limiting behavior as b -»> and reduces 
to (13.2) for b >b min . 

The lower limit on b can be obtained by another argument. Equation 
(13.2) was derived under the assumption that the electron did not move 
appreciably during the collision. As long as the distance d it actually 
moves is small compared to b, we may expect that (13.2) will be correct. 
An estimate of d can be obtained by saying that Ap/2m is an average 
velocity of the electron during the collision, and that the time of collision 
is given by (11.120). Hence the distance traveled during the collision is of 

the order of 

Ap . ze 2 

2m ymxr 

As long as b > d, (13.2) should hold. This is exactly the condition 
implied by (13.7). 

At the other extreme of very distant collisions the approximate result 
(13.2) for AE{b) is in error because of the binding of the atomic electrons. 
We assumed that the electrons were free, whereas they are actually bound 
in atoms. As long as the collision time (11.120) is short compared to the 
orbital period of motion, it may be expected that the collision will be sudden 
enough that the electron may be treated as free. If, on the other hand, the 
collision time (11.120) is very long compared to the orbital period, the 
electron will make many cycles of motion as the incident particle passes 
slowly by and will be influenced adiabatically by the fields with no net 
transfer of energy. The dividing point comes at impact parameter Z> max , 
where the collision time (1 1 . 120) and the orbital period are comparable. If 
co is a characteristic atomic frequency of motion, this condition is 



d~-P xAt = — % = b min (13.8) 



1 

At(b ma .x) ~ — 
co 
or 

b - y - 

o max — 

CO J 



(13.9) 



For impact parameters greater than & max it can be expected that the energy 
transfer falls below (13.2), going rapidly to zero for b > b max . 

The general behavior of AE(b) as a function of b is shown in Fig. 13.2. 
The dotted curve represents the approximate form (13.2), while the solid 



[Sect. 13.1] 



Collisions between Charged Particles 



433 



Fig. 13.2 Energy transfer as a 
function of impact parameter. 




log 6- 



curve is a representation of the correct result. In the interval b m iu < b < 
bm&x the energy transfer is given approximately by (13.2). But for impact 
parameters outside that interval, the energy transfer is considerably less. 
A fast particle passing through matter "sees" electrons at various 
distances from its path. If there are N atoms per unit volume with Z 
electrons per atom, the number of electrons located at impact parameters 
between b and (b + db) in a thickness dx of matter is 



dn = NZ 2irb db dx 



(13.10) 



To find the energy lost per unit distance by the incident particle we multiply 
(13.10) by the energy transfer AE(b) and integrate over all impact para- 
meters. Thus the energy loss is 

— = 2nNZ \AE(b)b db (13.11) 

dx J 

In view of the behavior of AE(b) shown in Fig. 13.2 we may use approxi- 
mation (13.2) and integrate between 6 m in and b mSbX . Then we find the result 



dE 
dx 



_2„4 /*bmax 1 

~4ttNZ— 9 - 9 bdb 

mv 2 Jbmin b l 



(13.12) 



434 



Classical Electrodynamics 



or 



dE zV 

— ~4ttNZ— JnB 



where 



dx 
B = 



mv 



y 2 mv z 
ze 2 co 



(13.13) 
(13.14) 



This approximate expression for the energy loss exhibits all the essential 
features of the classical result due to Bohr (1915). The method of handling 
the lower limit of integration in (13.12) is completely equivalent to using 
(13.7) for AE(b). The cutoff at b = b ma , x is only approximate. Con- 
sequently B is uncertain by a factor of the order of unity. Because B 
appears in the logarithm, this factor is of negligible importance numeri- 
cally. In any event, a proper treatment of binding effects is given in the 
next section. Discussion of (13.13) as a function of energy and its com- 
parison with experiment will be deferred until Section 13.3. 

13.2 Energy Transfer to a Harmonically Bound Charge 

In order to justify the plausible value Z? max (13.9) of the impact parameter 
which divides the Coulomb collisions for b < b max with the free-energy 
transfer (13.2) and essentially adiabatic collisions for b > b max with 
negligible energy transfer, we consider the problem of the energy lost by a 
massive charged particle with charge ze and velocity v passing a harmoni- 
cally bound charge of mass m and charge e. This will serve as a simplified 
model for energy loss of particles passing through matter. As before, we 
will assume that the massive particle is deflected only slightly in the 
encounter so that its path can be approximated by a straight line. It passes 
by the bound particle at an impact parameter b, measured from the origin 
O of the binding force, as shown in Fig. 13.3. Since we are primarily 
interested in large impact parameters where binding effects are important, 




[Sect. 13.2] Collisions between Charged Particles 435 

we may assume that the energy transfer is not large, that the motion of the 
bound particle is nonrelativistic throughout the collision, and that its 
initial and final amplitudes of oscillation about the origin O are small 
compared to b. Then only the electric field of the incident particle need 
be included in the force equation. Furthermore, its variation over the 
position of the bound particle may be neglected, and its effective value can 
be taken as that at the origin O. This is sometimes called the dipole approxi- 
mation, by analogy with the corresponding problem of absorption of 
radiation. 

With these approximations the force equation for the harmonically 
bound charge can be written as 

x + Tx + o> 2 x = - E(f) (13.15) 



m 



where E(0 is the electric field at O due to the charge ze, its components 
being given by (11.118), co is the characteristic frequency of the binding, 
and T is a small damping constant. The damping factor is not essential, 
but it is present to at least some degree in actual physical systems and serves 
to remove certain ambiguities which would arise in its absence. To solve 
(13.15) we Fourier-analyze both E(?) and x(t): 

x(0 = -L (*" x(co)e- i<ot dto (13.16) 

■J2TT J -oo 

1 f °° 
E(0 = -^= E((o)e- itot dco (13.17) 

■J2.1T J - oo 

Since both x(0 and E(f) are real, the positive and negative frequency parts 
of their transforms are related by 

x(— a>) = x*(co) 
V } V (13.18) 

E(-co) = E*(eo) 

When the Fourier integral forms are substituted into the force equation, 
we find 

*(»> = £ ^-^T ? (1319) 

m co — icol — a> 

With the known form of E(0 the Fourier amplitude E(co) can be deter- 
mined. Then x(0 can be found from (13.16), using (13.19). The problem 
is solved, provided one can do the integrals. 

The quantity of immediate interest is not the detailed motion of the 
bound particle, but the energy transfer in the collision. This can be found 
by considering the work done by the incident particle on the bound one. 



436 Classical Electrodynamics 

The rate of doing work is given by 

f=jE. Jd V ( 13 .20) 

Thus the total work done by the particle passing by is 

AE=j CO dt\d 3 x'E>J (13.21) 

The current density is J = ev d[x' — x(f)] for the bound charge. Con- 
sequently 

AE = e\ \>Edt (13.22) 

J — oo 

where v = x, and in the dipole approximation E is the field of the incident 
particle at the origin O. Using the Fourier representations (13.16) and 
(13.17), as well as that for a delta function (2.52), and the reality con- 
ditions (13.18), the energy transfer can be written 

AE = 2e Re -iwx(co) • E*(co) dco (13.23) 

If now the result (13.19) for x(co) is inserted, this becomes 

AE= e -\ |E(o>)| 2 1( °l d«> (13.24) 

m Jo (w 2 — co 2 ) 2 + coT 2 

For small T the integrand peaks sharply around a> = co in an approxi- 
mately Lorentzian line shape. Consequently the factor involving the 
electric field can be approximated by its value at co = co . Then (13.24) 
becomes 



AE = — |E(w )| 2 ' °° x%dx 



m 



co l- 3? 



o Lr 2 



(13.25) 



+ a; 2 



The integral has the value tt/2, independent of coJT. Thus the energy 
transfer is 

AE = ™ 2 |E(a> )| 2 (13.26) 



Equation (13.26) is a very general result for energy transfer to a non- 
relativistic oscillator by an external electromagnetic field. In the present 
application the field is produced by a passing charged particle. But a 
pulse of radiation or any combination of external fields will serve as well. 
For a particle with charge ze passing by the origin O at an impact 
parameter b with a velocity v, the electromagnetic fields at the origin are 



[Sect. 13.2] 



Collisions between Charged Particles 



437 



given by (11.118) with q = ze. To illustrate the determination of the 
Fourier transform (1 3. 1 7) we consider E x (t). Its transform E^w) is defined 
to be 

^^^rJ-^+ywr (13 - 27) 

By changing integration variable to x = yvtjb, (13.27) can be written as 

-iabxjyv 



£l(«>) = 



ze 



r 



dx 



(13.28) 



From a table of Fourier transforms* we find that the integral is propor- 
tional to a modified Bessel function of the order of unity [see (3.101)]. 
Thus 

£i(C0)= 5£.(2fr^ Xi (^)i (13 . 29) 

bv\TT/ Lyv \yv'- 
Similarly E 3 (t) given by (11.118) has the Fourier transform: 

£3(<u )=_,-il(2fr^K (^)l (13.30) 



The energy transfer (13.26) to the harmonically bound charge can now 
be evaluated explicitly. Using (13.29) and (13.30), we find 



AE(b) = 



2zV 



& 



where 



y 2 



0) n b 



yv 



(13.31) 



(13.32) 



The factor multiplying the square bracket is just the approximate result 
(13.2). For small and large i, the limiting forms (3.103) and (3.104) show 
that the square bracket in (13.31) has the limiting values: 



[ ] = 



1, 



iK)i 



!e 



2| 



for £ < 1 
for £ > 1 



(13.33) 



Since £ = bjb m&x , we see that for b < b m&x the energy transfer is essentially 
the approximate result (13.2), while for b > Z> max it falls off exponentially 
to zero. This justifies the qualitative arguments of the previous section on 
the upper limit 6 m ax- 

* See, for example, Magnus and Oberhettinger, Chapter VIII, or Bateman Manuscript 
Project, Tables of Integral Transforms, Vol. I, Chapters I— III. 



438 Classical Electrodynamics 



13.3 Classical and Quantum-Mechanical Energy-Loss Formulas 

The energy transfer (13.31) to a harmonically bound charge can be used to 
calculate a classical energy loss per unit length for a fast, heavy particle 
passing through matter. We suppose that there are N atoms per unit 
volume with Z electrons per atom. The Z electrons can be divided into 
groups specified by the indexy, with^ electrons having the same harmonic 
binding frequency co } . The number f j is called the oscillator strength of 
they'th oscillator. The oscillator strengths satisfy the obvious sum rule, 
2/; — z - Bv a trivial extension of the arguments leading to (13.11) 

3 

and (13.12) we find the energy loss to be 



dE ^Z* f °° 

— = 2ttN 2, /, kE^b db (13.34) 

dX j J bmin 

where A£/6) is given by (13.31) with I = cofi/yv, and a lower limit of 
6min is specified, consistent with (13.7). No upper limit is necessary, since 
(13.31) falls rapidly to zero for large b. The integral over the modified 
Bessel functions can be done in closed form, leading to the result, 



dE_ 47rNz 2 e* 
dx mv 2 i 



v 2 

£min-Ki(! : min).K'o(£min)— — „ lmin(^ 1 2 (f min ) — K 2 (£ m in)) 

2c 

(13.35) 

where | min = wfiminlyv. In general, £ mi n < 1 . This means that the 
limiting forms (3. 103) may be used to simplify (13.35). This final expression 
for classical energy loss is 



^ = 4nNZ — 
dx mv 2 



. ° 2c 2 J 



(13.36) 



where the argument of the logarithm is 



_ \.\23yv 1.123yW 
Be ~ 7TI — = i^^ — (13.37) 

The average frequency (co) appearing in B c is a geometric mean defined 
by 

Z In (co) = 2L In co } (13.38) 

3 

The result (13.36)— (13.38) is that obtained by Bohr in his classic paper on 
energy loss (1915). Our approximate expression (13.13) is in agreement 
with (13.36) in all its essentials, since the added —v 2 /2c 2 is a small cor- 
rection even at high velocities. 



[Sect. 13.3] Collisions between Charged Particles 439 

Bohr's formula (13.36) gives a reasonable description of the energy loss 
of relatively slow alpha particles and heavier nuclei. But for electrons, 
mesons, protons, and even fast alphas, it overestimates the energy loss 
considerably. The reason is that for the lighter particles quantum- 
mechanical modifications cause a breakdown of the classical result. The 
important quantum effects are (1) discreteness of the possible energy 
transfers, and (2) limitations due to the wave nature of the particles and 
the uncertainty principle. 

The problem of the discrete nature of the energy transfer can be illus- 
trated by calculating the classical energy transfer (13.2) at b ~ & m ax. This 
is roughly the smallest energy transfer that is of importance in the energy- 
loss process. Assuming only one binding frequency a> for simplicity, we 
find 

AE(fr max ) ~ - 2 z 2 (ffi™o (13-39) 

where v = c/137 is the orbital velocity of an electron in the ground state 
of hydrogen. Since Hco is of the order of the ionization potential of the 
atom, we see that for a fast particle (v > v ) the classical energy transfer 
(13.39) is very small compared to the ionization potential, or even to the 
smallest excitation energy in the atom. But we know that energy must be 
transferred in definite quantum jumps. A tiny amount of energy like (1 3.39) 
simply cannot be absorbed by the atom. We conclude that our classical 
calculation fails in this domain. We might argue that only if our classical 
formula (13.2) gives an energy transfer large compared to typical atomic 
excitation energies would we expect it to be correct. This would set quite 
a different upper limit on the impact parameters. Fortunately the classical 
result can be applied in a statistical sense if we reinterpret its meaning. 
Quantum considerations show that the classical result of the transfer of a 
small amount of energy in every collision is incorrect. But if we consider 
a large number of collisions, we find that on the average a small amount 
of energy is transferred. It is not transferred in every collision, however. 
In most collisions no energy is transferred. But in a few collisions an 
appreciable excitation occurs, yielding a small average value over many 
collisions. In this statistical sense the quantum mechanism for discrete 
energy transfers and the classical process with a continuum of possible 
energy transfers can be reconciled. The detailed numerical agreement 
stems from the quantum-mechanical definitions of the oscillator strengths 
fj and resonant frequencies co>. 

The other important quantum modification arises from the wave nature 
of the particles. The uncertainty principle sets certain limits on the range 
of validity of classical orbit considerations. If we try to construct wave 



440 Classical Electrodynamics 

packets to give approximate meaning to a classical trajectory, we know 
that the path can be defined only to within an uncertainty Ax > hjp. For 
impact parameters b less than this uncertainty, classical concepts fail. 
Since the wave nature of the particles implies a smearing out in some sense 
over distances of the order of Ax, we anticipate that the correct quantum- 
mechanical energy loss will correspond to much smaller energy transfers 
than given by (13.2) for b < Ax. Thus Ax ~ hip is a quantum analog of 
the minimum impact parameter (13.6). In the collision of two particles 
each one has a wave nature. For a given relative velocity the limiting 
uncertainty will come from the lighter of the two. For a heavy incident 
particle colliding with an electron, the momentum of the electron in the 
coordinate frame where the incident particle is at rest (almost the CM 
frame) is p' = ymv, where m is the mass of the electron. Therefore the 
quantum-mechanical minimum impact parameter is 

h 

Cn = (13.40) 

ymv 

For electrons incident on electrons we must take more care and consider 
the CM momentum (12.34) for equal masses. Then for electrons we obtain 
the minimum impact parameter, 

(«) -\ ^ I ^ 

[^min J electrons = / (13.41) 

mc ^ y — 1 

In a given situation the larger of the two minimum impact parameters 
(13.6) and (13.40) must be used to define argument B (13.14) of the 
logarithm in dE/dx. The ratio of the classical to quantum value of b min is 

ze 2 
ri = — (13.42) 

nv 

Ift]> 1, the classical Bohr formula must be used. We see that this occurs 
for slow, highly charged, incident particles, in accord with observation. 
If r\ < 1, the quantum minimum impact parameter is larger than the 
classical one. Then quantum modifications appear in the energy-loss 
formula. The argument of the logarithm in (13.13) becomes 

„ bmax „ y 2 mv 2 

B « = -^r =r i B = -r-7 (13.43) 

Equation (13.13) with the quantum-mechanical B q (13.43) in the logarithm 
is a good approximation to a quantum-theoretical result of Bethe (1930). 
Bethe's formula, including the effects of close collisions, is 



dE z 2 e* 

^ = 4ttNZ — 
dx mv 2 



ln 1 2 r 2mv '' 



h{<o) 



J c\ 



(13.44) 



[Sect. 13.3] 



Collisions between Charged Particles 



441 



Apart from the small correction term -v 2 \c 2 and a factor of 2 in the 
argument of the logarithm, this is just our approximate expression. 

For electrons the quantum effects embodied in (13.41) lead to a modified 
quantum-mechanical argument for the logarithm: 



B d ^ (y - 1) 



y A mc 2 



j y + 1 mc 2 

2 hico)^ ~^2 h(a>) 



(13.45) 



where the last expression is valid at high energies. Even though there are 
other quantum effects for electrons, such as spin and exchange effects, the 
dominant modifications are included in (13.45). 

The general behavior of both the classical and quantum-mechanical 
energy-loss formulas is shown in Fig. 13.4. At low energies, the main 
energy variation is as v~ 2 , since the logarithm changes slowly. But at high 
energies where v -► c the variation is upwards again, going as In y for 
y > 1. Bethe's formula is in good agreement with experiment for all fast 
particles with r\ < 1, provided the energy is not too high (see the next 
section). 

It is worth while to note the physical origins of the two powers of y 
which appear in B Q (13.43). One power of y comes from the increase of 
the maximum energy (13.5) which can be transferred in a head-on collision. 
The other power comes from the relativistic change in shape of the electro- 
magnetic fields (11.118) of a fast particle with the consequent shortening 
of the collision time (11.120) and increase of fcmax (13.9). The fields are 
effective in transferring energy at larger distances for a relativistic particle 
than for a nonrelativistic one. 

Sometimes it is of interest to know the energy loss per unit distance due 
to collisions in which less than some definite amount e of energy is trans- 
ferred per collision. In photographic emulsions, for example, ejected 



Fig. 13.4 Energy loss as a 
function of kinetic energy. 



0.01 0.1 



1 



10 



HT 



10" 5 



10 4 



<T- 1)= ^ 



442 Classical Electrodynamics 

electrons of more than about 10 Kev energy have a range greater than the 
average linear dimensions of the silver bromide grains. Consequently the 
energy dissipated in blackening of the grains corresponds to collisions 
where the energy transfer is less than about 10 Kev. Classically, the 
desired energy-loss formula can be obtained from the Bohr formula (13.35) 
with a minimum impact parameter b min (€) chosen so that (13.2) is equal 
to e. Thus 

2ze 2 
&min(e) = — — T (13.46) 

u(2rae) 

This leads to a formula of the form of (13.36), but with an argument in 
the logarithm, 

B/O- '-T* 2 "*' (1347) 

Since quantum-mechanical energy-loss formulas are obtained from clas- 
sical ones by the replacement [see (13.43)], 

ze 2 
B a = rjB c = — B c (13.48) 

nv 

we expect that the quantum-mechanical formula for energy-loss per unit 
distance due to collisions with energy transfer less than e will be 



(13.49) 



dF 2 V r Ji 2 

f*( € ) = 4 7 rNZ^ 2 \nB Q (e)-^ 
ax mtr L 2c - 

where , , 

BJA-XV&&- (13.50) 

n{co) 

The constant A is a numerical factor of the order of unity that cannot be 
determined without detailed quantum-mechanical calculations. Bethe's 
calculations (1930) give the value A = 1. The quantum-mechanical 5 g (e) 
can be written as 

where 6 max is given by (13.9), and the minimum impact parameter is 

b$ n (e)~—~——^ (13.52) 

The implication of this formula is that the classical trajectory must be ill 
defined by an amount at least as great as (13.52) in order that the uncertainty 
in transverse momentum Ap be less than the momentum transfer in the 
collision. Otherwise we would be unable to be certain that an energy 
transfer of less than e had actually occurred. Hence (13.52) forms a 
natural quantum-mechanical lower limit on the classical orbit picture in 
this case. 



[Sect. 13.4] Collisions between Charged Particles 443 

13.4 Density Effect in Collision Energy Loss 

For particles which are not too relativistic the observed energy loss is 
given accurately by (13.44) [or by (13.36) if rj > 1] for all kinds of particles 
in all types of media. For ultrarelativistic particles, however, the observed 
energy loss is less than predicted by (13.44), especially for dense substances. 
In terms of Fig. 13.4 of (dE/dx), the observed energy loss increases beyond 
the minimum with a slope of roughly one-half that of the theoretical curve, 
corresponding to only one power of y in the argument of the logarithm 
in (13.44) instead of two. In photographic emulsions the energy loss, as 
measured from grain densities, barely increases above the minimum to a 
plateau extending to the highest known energies. This again corresponds 
to a reduction of one power of y, this time in ^(e) (13.50). 

This reduction in energy loss, knbwn as the density effect, was first 
treated theoretically by Fermi (1940). In our discussion so far we have 
tacitly made one assumption that is not valid in dense substances. We 
have assumed that it is legitimate to calculate the effect of the incident 
particle's fields on one electron in one atom at a time, and then sum up 
incoherently the energy transfers to all the electrons in all the atoms with 
b min < * < &max. Now & max is very>rge compared to atomic dimensions, 
especially for large y. Consequently in dense media there are many atoms 
lying between the incident particle's trajectory and the typical atom in 
question if b is comparable to £ ma x. These atoms, influenced themselves 
by the fast particle's fields, will produce perturbing fields at the chosen 
atom's position, modifying its response to the fields of the fast particle. 
Said in another way, in dense media the dielectric polarization of the 
material alters the particle's fields from their free-space values to those 
characteristic of macroscopic fields in a dielectric. This modification of 
the fields due to polarization of the medium must be taken into account 
in calculating the energy transferred in distant collisions. For close 
collisions the incident particle interacts with only one atom at a time. Then 
the free-particle calculation without polarization effects will apply. The 
dividing impact parameter between close and distant collisions is of the 
order of atomic dimensions. Since the joining of two logarithms is involved 
in calculating the sum, the dividing value of b need not be specified with 
great precision. 

We will determine the energy loss in distant collisions (b > a), assuming 
that the fields in the medium can be calculated in the continuum approxi- 
mation of a macroscopic dielectric constant e(co). If a is of the order of 
atomic dimensions, this approximation will not be good for the closest of 
the distant collisions, but will be valid for the great bulk of the collisions. 



444 



Classical Electrodynamics 



The problem of finding the electric field in the medium due to the incident 
fast particle moving with constant velocity can be solved most readily by 
Fourier transforms. If the potentials A^x) and source density J (x) are 
transformed in space and time according to the general rule, 



x —iwt 



(13.53) 



(13.54) 



F(x, t) = — 2 \d*k \dco F(k, co)e ik ' 

then the transformed wave equations become 

m 2 1 4tt "• 

k 2 --e(co) 0>(k, co) = -^- p(k, co) 
L c L J e(co) 

k 2 -^- e(co) 1 A(k, co) = — J(k, co) 
L c 2 J c 

The dielectric constant e(co) appears characteristically in positions dictated 
by the presence of D in Maxwell's equations. The Fourier transforms of 

p(x, t) = zed(x - Yt)) 
and (13.55) 

J(x, = \ P (x, t) J 

are readily found to be 



ze 



/>(k, co) = — d(co — k • v) 

Itt 



(13.56) 



J(k, a>) = vp(k, oi) 
From (13.54) we see that the Fourier transforms of the potentials are 

2ze d(co — k • v) 



0(k, co) = 



and 



e(co) , 2 co 2 . 
c 2 



A(k, co) = e(oj) - 0(k, co) 



(13.57) 



From the definitions of the electromagnetic fields in terms of the potentials 
we obtain their Fourier transforms : 



_,,, x .((0€(co)y . ,^,, 
E(k, co) = / — L - '- - — k )^>(k, co) 

\ c c 



B(k, co) = ie(co) k x - <D(k, co) 
c 



(13.58) 



In calculating the energy loss it is apparent from (13.23) that we want 
the Fourier transform in time of the electromagnetic fields at a perpen- 
dicular distance b from the path of the particle moving along the z axis. 
Thus the required electric field is 



E(co) = — J-yr d 3 k E(k, co) e ibkl 
(2tt)- 2 J 



(13.59) 



[Sect. 13.4] Collisions between Charged Particles 445 

where the observation point has coordinates (b, 0, 0). To illustrate the 
determination of E(w) we consider the calculation of E 3 (w), the component 
of E parallel to v. Inserting the explicit forms from (13.57) and (13.58), we 
obtain 

«„) = ^ > U ,- (^ - is )ii^>- (13.60) 



/c 2 -^-<o>) 
c 2 



The integral over d£: 3 can be done immediately. Then 

(13.61) 

where 22 2 

A 2 = «» _ £L e(co) = ^ (1 - £ 2 <co)) (13.62) 

y 2 c 2 U 

The integral over dk % has the value 77/(A 2 + ^i 2 ) 1 ^, so that E z {oS) can be 
written 

«->--^&-')£<F^ (13 - 63) 

The remaining integral is of the same general structure as (13.28). The 
result is 

£ 3(K1 ) = - J22L (2) M (J- - /)•) K (A!>) (13.64) 

where the square root of (13.62) is chosen so that A lies in the fourth 
quadrant. A similar calculation yields the other fields : 

X 



V \tt/ 



K&by 



6(0) (13.65) 

B 2 (co) = e(co) / S£ 1 (co) > 

In the limit e(co) -> 1 it is easily seen that fields (13.64) and (13.65) reduce 
to the earlier results (13.30) and (13.29). 

To find the energy transferred to the atom at impact parameter b we 
merely write down the generalization of (13.23): 

J"QO 
- io>x/co) . E*(o>) dm (13.66) 


where ^{co) is the amplitude of the yth type of electron in the atom. 
Rather than use (13.19) for x/cu) we express the sum of dipole moments 
in terms of the molecular polarizability and so the dielectric constant: 

e Y/,x» = — (e(co) - l)E(co) (13.67) 

■£-< 4ttN 



446 Classical Electrodynamics 

where TV is the number of atoms per unit volume. Then the energy transfer 
can be written 

1 f 00 

AE(b) = — - Re -ia> e(co) |E(co)| 2 dco (13.68) 

zttN Jo 

The energy loss per unit distance in collisions with impact parameter 
b > a is evidently 

(fL = HT A£(fe > wi < i3 - 69 > 

If fields (13.64) and (13.65) are inserted into (13.68) and (13.69), we find, 
after some calculation, the expression due to Fermi, 

\t\ = - ( -nr Re | ™&°Ki&*a)KMa)(-r--p\dm (13.70) 

\dx/b>a 77- v* Jo Ve(ft)) / 

where X is given by (13.62). This result can be obtained more elegantly by 
calculating the electromagnetic energy radiated through a cylinder of 
radius a around the path of the incident particle. By conservation of 
energy this is the energy lost per unit time by the incident particle. Thus 

(dE\ IdE c r „ 

— I =--T=-T— 2naB 2 E 3 dz (13.71) 

\dx/b>a v dt ^vJ-oo ' v ' 

The integral over dz at one instant of time is equivalent to an integral at 
one point on the cylinder over all time. Using dz = v dt, we have 

(Si>.--f r.^^ * (i372) 

In the standard way this can be converted into a frequency integral, 

— ) = -ca Re B 2 *(<o)E 3 (o)) da> (13.73) 

\dx/b>a J 

With fields (13.64) and (13.65) this gives the Fermi result (13.70). 

The Fermi expression (13.70) bears little resemblance to our previous 
results for energy loss, such as (13,35). But under conditions where 
polarization effects are unimportant it yields the same results as before. 
For example, for nonrelativistic particles (/ff < 1) it is clear from (13.62) 
that X ^ co/v, independent of e(o>). Then in (13.70) the modified Bessel 
functions are real. Only the imaginary part of l/e(co) contributes to the 
integral. If we neglect the Lorentz polarization correction (4.67) to the 
internal field at an atom, the dielectric constant can be written 

«■>* 1+^2 . S i , r (13.74) 



[Sect. 13.4] Collisions between Charged Particles 447 

where we have used the dipole moment expression (13.19). Assuming 
that the second term is small, the imaginary part of l/e(co) can be readily 
calculated and substituted into (13.70). Then the integral over dco can be 
performed in the same approximation as used in (13.24)— (13.26) to yield 
the nonrelativistic form of (13.35). If the departure of A from cojyv is 
neglected, but no other approximations are made, then (13.70) yields 
precisely the Bohr result (13.35). 

The density effect evidently comes from the presence of complex 
arguments in the modified Bessel functions, corresponding to taking into 
account e(co) in (13.62). Since e(co) there is multiplied by /5 2 , it is clear that 
the density effect can be really important only at high energies. The 
detailed calculations for all energies with some explicit expression such as 
(13.74) for e(co) are quite complicated and not particularly informative. 
We will content ourselves with the extreme relativistic limit (/? ~ 1). 
Furthermore, since the important frequencies in the integral over da> are 
optical frequencies and the radius a is of the order of atomic dimensions, 
\Xa\ ^- ' (coa/c) < 1 . Consequently we can approximate the Bessel functions 
by their small argument limits (3.103). Then in the relativistic limit the 
Fermi expression (13.70) is 



\dx/b>a 77 c 2 Jo Wco) / 



[.(! 



(CO) 

?V\ 1 1 

dw (13.75) 



123c \ 1 

In (1 — e(a>)) 



coa J 2 

It is worth while right here to point out that the argument of the second 
logarithm is actually [1 — /# 2 e(co)]. In the limit e = 1, this log term gives 
a factor y in the combined logarithm, corresponding to the old result 
(13.36). Provided e(co) =£ 1, we can write this factor as [1 — €(co)], 
thereby removing one power of y from the logarithm, in agreement with 
experiment. 

The integral in (13.75) with e(co) given by (13.74) can be performed most 
easily by using Cauchy's theorem to change the integral over positive real 
o> to one over positive imaginary co, minus one over a quarter circle at 
infinity. The integral along the imaginary axis gives no contribution. Pro- 
vided the Tj in (13.74) are assumed constant, the result of the integration 
over the quarter circle can be written in the simple form: 

\dx/b>a c £ \ aco p I 

where co„ is the electronic plasma frequency 

, 4nNZe 2 



m 



(13.77) 



448 Classical Electrodynamics 

The corresponding relativistic expression without the density effect is, 

from (13.36), 

(dE\ = (zefail T /U23yc\ _ l\ 
\dx/b>a c 2 L \ a(co) J 2J 

We see that the density effect produces a simplification in that the 
asymptotic energy loss no longer depends on the details of atomic 
structure through (a>) (13.38), but only on the number of electrons per unit 
volume through co p . Two substances having very different atomic struc- 
tures will produce the same energy loss for ultrarelativistic particles pro- 
vided their densities are such that the density of electrons is the same in each. 
Since there are numerous calculated curves of energy loss based on 
Bethe's formula (13.44), it is often convenient to tabulate the decrease in 
energy loss due to the density effect. This is just the difference between 
(13.78) and (13.76): 



P-+i \dxf c 2 L \<o>>/ 2 



(13.79) 



For photographic emulsions, the relevant energy loss is given by (13.49) 
and (13.50) with e^ 10 Kev. With the density correction applied, this 
becomes constant at high energies with the value, 



2_ 2 



dE(e) (zeYco P 



dx 2c 2 



/2mc 2 e \ 
U 2 co 2 / 



In \^i) (13.80) 



For silver bromide, hco p ^ 48 ev. Then for singly charged particles (13.80), 
divided by the density, has the value of approximately 1 .02 Mev-cm 2 /gm. 
This energy loss is in good agreement with experiment, and corresponds 
to an increase above the minimum value of less than 10 per cent. Figure 
13.5 shows total energy loss and loss from transfers of less than 10 Kev 
for a typical substance. The dotted curve is the Bethe curve for total 
energy loss without correction for density effect. 

There is an interesting connection between the Fermi expression (13.70) 
for energy loss and the emission of Cherenkov radiation. Equation (13.70) 
represents energy transferred to the medium at distances greater than a. 
If we let a -> oo we can find out whether any of the energy escapes to 
infinity. Such energy would be properly described as radiation. For 
a -> oo, the asymptotic forms (3.104) of the ^functions can be used. Then 
(13.70) takes the form: 

lira («) _ & Re r- to (j_ _ ^(m-<— >» *, (13.8D 

a— oo \dx/b>a \) Jq \e(<X>) I \ A. / 



[Sect. 13.4] 



Collisions between Charged Particles 



449 



31-3 




< 10 kev 



0.1 



10 10 2 10 3 

(T - 1) ^ 



10 4 



Fig. 13.5 Energy loss, including the density effect. The dotted curve is the total energy 

loss without density correction. The solid curves have the density effect incorporated, 

the upper one being the total energy loss and the lower one the energy loss due to 

individual energy transfers of less than 10 Kev. 



If A has a real part, the exponential factor causes the energy loss to go 
rapidly to zero at large distances. From (13.62) it is evident that this will 
always occur if the medium is absorbent, since then e(o>) has a positive 
imaginary part. But if e(w) is real, A can be pure imaginary for certain co. 
This occurs whenever ft* > l/e(co), i.e., whenever the velocity of the part- 
icle is greater than the phase velocity of light in the medium. This is the 
criterion for Cherenkov radiation. For such frequencies, A = — * |A|. 
Then the exponential equals unity, and we find 

Um (M) _6£f Jt- _J_ ) dm (13 .82) 

Since this expression is independent of the cylinder radius a, it represents 
true radiation. It is just the Frank-Tamm (1937) result for the total energy 
per unit distance emitted as Cherenkov radiation. A more detailed 
discussion of Cherenkov radiation as a radiative process will be given in 
Section 14.9. 

For media in which the density effect is an important feature of the 
energy-loss process the absorption is almost always sufficiently great that 
the incipient Cherenkov radiation is absorbed very close to the path of the 
particle. 



450 Classical Electrodynamics 

13.5 Energy Loss in an Electronic Plasma 

The loss of energy by a nonrelativistic particle passing through a plasma 
can be treated in a manner similar to the density effect for a relativistic 
particle. As was discussed in Section 10.10, the length scale in a plasma is 
divided into two regions. For dimensions large compared to the Debye 
screening distance k D ~ x (10.106), the plasma acts as a continuous medium 
in which the charged particles participate in collective behavior such as 
plasma oscillations. For dimensions small compared to k D ~ x , individual- 
particle behavior dominates and the particles interact by the two-body 
screened potential (10.1 13). This means that in calculating energy loss the 
Debye screening distance plays the same role here as the atomic dimension 
a played in the density-effect calculation. For close collisions collective 
effects can be ignored, and the two-body screened potential can be used to 
evaluate this contribution to the energy loss. This is left as an exercise for 
the reader (Problem 13.3). For the distant collisions at impact parameters 
bk D > 1 the collective effects can be calculated by utilizing Fermi's 
formula (13.70) with an appropriate dielectric constant for a plasma. The 
loss in distant collisions corresponds to the excitation of plasma oscillations 
in the medium. 

For a nonrelativistic particle (13.70) yields the following expression for 
the energy loss to distances b > k D ~ x : 



*l\ ~ 2 (* g ) 2 

dx'k s b>l 77 V 2 



Jo €(oj)\-k D v \k D v/ \k D v/ . 



dm (13.83) 



Since the important frequencies in the integral turn out to be m ~ m v , the 
relevant argument of the Bessel functions is 



m v _ {u ) 
k n v v 



(13.84) 



For particles incident with velocities v less than thermal velocities this 
argument is large compared to unity. Because of the exponential fall-off 
of the Bessel functions for large argument, the energy loss in exciting 
plasma oscillations by such particles is negligible. Whatever energy is lost 
is in close binary collisions. If the velocity is comparable with or greater 
than thermal speeds, then the particle can lose appreciable amounts of 
energy in exciting collective oscillations. It is evident that this energy of 
oscillation is deposited in the neighborhood of the path of the particle, 
out to distances of the order of (v/(u 2 y A ) kjf x . 



[Sect. 13.6] Collisions between Charged Particles 451 

For a particle moving rapidly compared to thermal speeds we may use 
the familiar small argument forms for the modified Bessel functions. Then 
(13.83) becomes 

(f) = ^r Re /^,/u23M) dM (1385) 

\ax>ic D b>i it v Jo \e(ft>)/ \ co 1 

We shall take the simple dielectric constant (7.93), augmented by some 
damping: 

<°* = 1 ~ a?' T, ( 13 - 86 ) 

co 1 + icoT 

The damping constant Y will be assumed small compared to co v . The 
necessary combination, 

Re (fl) = < rs %+ v> (I3 ' 87) 

\e(a))/ (co — co v ) + co L 

has the standard resonant character seen in (13.24), for example. In the 
limit r < co v the integral in (13.85) leads to the simple result, 

Ui\ ^ a/la (l^hnE) (1 3.88) 

This can be combined with the results of Problem 13.3 to give an expression 
for the total energy loss of a particle passing through a plasma. The 
presence of co v in the logarithm implies that the energy losses occur in 
quantum jumps of Hco p , in the same way as the mean frequency (co) in 
(13.44) is indicative of the typical quantum jumps in atoms. Electrons 
passing through thin metal foils show this discreteness in their energy loss. 
The phenomenon can be used to determine the effective plasma frequency 
in metals. 



13.6 Elastic Scattering of Fast Particles by Atoms 

In the preceding sections we have been concerned with the energy loss 
of particles passing through matter. In these considerations it was assumed 
that the trajectory of the particle was a straight line. Actually this 
approximation is not rigorously true. As was discussed in Section 13.1, 
any momentum transfer between collision partners leads to a deflection in 
angle. In the introductory remarks at the beginning of the chapter it was 
pointed out that collisions with electrons determine the energy loss, 
whereas collisions with atoms determine the scattering. If the screening of 
the nuclear Coulomb field by the atomic electrons is neglected, a fast 



452 Classical Electrodynamics 

particle of momentum/? = yMv and charge ze, passing a heavy nucleus of 
charge Ze at impact parameter b, will suffer an angular deflection, 

2z7e 2 
~ ±f±L (13.89) 

pvb 
according to (13.3). 

The differentia] scattering cross section da/dQ. (with dimensions of area 
per unit solid angle per atom) is defined by the relation, 

nbdbd<f> = n — sin 6 d6 d<f> (13.90) 

dQ. 

where n is the number of particles incident on the atom per unit area per 
unit time. The left-hand side of (1 3.90) is the number of particles per unit 
time incident at azimuthal angles between cf> and ((f> + d(f>) and impact 
parameters between b and (b + db). The right-hand side is the number of 
scattered particles per unit time emerging at polar angles (6, (f>) in the 
element of solid angle d£l = sin 6 dd d<f>. Equation (13.90) is merely a 
statement of conservation of particles, since b and 6 are functionally 
related. The classical differential scattering cross section can therefore be 
written. 



da 



dQ. sin 



db 
dd 



(13.91) 



The absolute value sign is put on, since db and dd can in general have 
opposite signs, but the cross section is by definition positive definite. If b 
is a multiple-valued function of 6, then the different contributions must be 
added in (13.91). 

With relation (13.89) between b and 6 we find the small-angle nuclear 
Rutherford scattering cross section per atom, 

*^(™£\ % L (13 .92) 

dQ \ pv I 4 

We note that the Z electrons in each atom give a contribution Z -1 times 
the nuclear one. Hence the electrons can be ignored, except for their 
screening action. The small-angle Rutherford law (13.92) for nuclear 
scattering is found to be true quantum mechanically, independent of the 
spin nature of the incident particles. At wide angles spin effects enter, but 
for nonrelativistic particles the classical Rutherford formula, 

^ = (^£!Y cosec ^ (13.93) 

dQ \2MvV 2 

which follows from (13.4), holds quantum mechanically as well. 



[Sect. 13.6] Collisions between Charged Particles 453 

Since most of the scattering occurs for 6 < 1, and even at 6 = tt\2 the 
small-angle result (13.92) is within 30 per cent of the Rutherford expression, 
it is sufficiently accurate to employ (13.92) at all angles for which the 
unscreened point Coulomb-field description is valid. 

Departures from the point Coulomb-field approximation come at large 
and small angles, corresponding to small and large impact parameters. At 
large b the screening effects of the atomic electrons cause the potential to 
fall off more rapidly than (1/r). On the Fermi-Thomas model the potential 
can be approximated roughly by the form : 

V(r) ~ -^- exp (-r/a) (13.94) 

r 

where the atomic radius a is 

a^l.4«oZ- 1/3 (13.95) 

The length a = h 2 /me 2 is the hydrogenic Bohr radius. For impact param- 
eters of the order of, or greater than, a the rapid decrease of the potential 
(13.94) will cause the scattering angle to vanish much more rapidly with 
increasing b than is given by (13.89). This implies that the scattering cross 
section will flatten off at small angles to a finite value at 6 = 0, rather than 
increasing as -4 . A simple calculation with a cutoff Coulomb potential 
shows that the cross section has the general form: 

f^f^T l (13 96) 

dQ. \ P V / (0 2 + 0Ln) 2 

where min is a cutoff angle. The minimum angle m in below which the 
cross section departs appreciably from the simple result (13.92) can be 
determined either classically or quantum mechanically. As with b m i n in 
the energy-loss calculations, the larger of the two angles is the correct one 
to employ. Classically min can be estimated by putting b = a in (13.89). 
This gives 

0^n~— (13.97) 

pva 

Quantum mechanically, the finite size of the scatterer implies that the 
approximately classical trajectory must be localized to within Aa; < a; 
the incident particle must have a minimum uncertainty in transverse 
momentum Ap ^ h/a. For collisions in which the momentum transfer 
(13.1) is large compared to H/a the classical Rutherford formula will apply. 
But for smaller momentum transfers we expect the quantum-mechanical 
smearing out to flatten off the cross section. This leads to a quantum 
mechanical d m i n : 

( X ^ — (13.98) 

pa 



454 Classical Electrodynamics 

We note that the ratio of the classical to quantum-mechanical angles 
0min is Zze 2 /hv in agreement with the ratio (13.42) of the classical and 
quantum values of b m in. For fast particles in all but the highest Z sub- 
stances (Zze 2 lhv) is less than unity. Then the quantum value (13.98) will 
be used for min . With value (13.95) for the screening radius a, (13.98) 
becomes 

ft^H (13.99) 

192 \ p I 

where p is the incident momentum (p = yMv), and m is the electronic 
mass. 

At comparatively large angles the cross section departs from (13.92) 
because of the finite size of the nucleus. For electrons and mu mesons the 
influence of nuclear size is a purely electromagnetic effect, but for pi mesons 
protons, etc., there are specific effects of a nuclear-force nature as well. 
Since the gross overall effect is to lower the cross section below that 
predicted by (13.92) for whatever reason, we will consider only the 
electromagnetic aspect. The charge distribution of the atomic nucleus can 
be crudely approximated by a uniform volume distribution inside a sphere 
of radius R, falling rapidly to zero outside R. This means that the electro- 
static potential inside the nucleus is not 1/r, but rather parabolic in shape 
with a finite value at r = 0: 



2 R \ 3R 2 ' 



V(r) = 



zZe 2 



(13.100) 
for r > R 



It is a peculiarity of the point-charge Coulomb field that the quantum- 
mechanical cross section is the classical Rutherford formula. Thus for a 
point nucleus there is no need to consider a division of the angular region 
into angles corresponding to impact parameters less than, or greater than, 
the quantum-mechanical impact parameter b<£ in (13.40). For a nucleus 
of finite size, however, the de Broglie wavelength of the incident particle 
does enter. When we consider wave packets incident on the relatively 
constant (inside r = R) potential (13.100), there will be appreciable 
departures from the simple formula (13.92). The situation is quite 
analogous to the diffraction of waves by a spherical object, considered in 
Chapter 9. The scattering is all confined to angles less than ~(X/R), where 
X is the wavelength (divided by 2tt) of the waves involved. For wider 
angles the wavelets from different parts of the scatterer interfere, causing 
a rapid decrease in the scattering or perhaps subsidiary maxima and 



[Sect. 13.6] 



Collisions between Charged Particles 



455 



Fig. 13.6 Atomic scattering, 
including effects of electronic 
screening at small angles and 
finite nuclear size at large angles. 




Iog0- 



minima. Since the particle wavelength is X = h/p, the maximum scattering 
angle, beyond which the scattering cross section falls significantly below the 
0~ 4 law, is 



pR 



(13.101) 



Using the simple estimate R ~ % (e 2 /mc 2 ) A 1A = \AA V * x 10 -13 cm, this 
has the numerical value, 

/ / /I / wi /> v 

(13.102) 



274 



mc 



A" \p 

We note that, for all values of Z and A, 6 ma , x > m i n . If the incident 
momentum is so small that max > 1 , the nuclear size has no appreciable 
effect on the scattering. For an aluminum target max = 1 when p ~ 50 
Mev/c, corresponding to ^ '50 Mev, 12 Mev, and 1.3 Mev kinetic energies 
for electrons, mu mesons, and protons, respectively. Only at higher 
energies than these are nuclear-size effects important in the scattering. At 
this momentum value d$ in ~ 10 -4 radian. 

The general behavior of the cross section is shown in Fig. 13.6. The 
dotted curve is the small-angle Rutherford approximation (13.92), while 
the solid curve shows the qualitative behavior of the cross section, includ- 
ing screening and finite nuclear size. The total scattering cross section can 
be obtained by integrating (13.96) over all solid angle: 

• fl^fl^^o /2zZe 2 \ 2 f°° Odd 
sin 6 dd d(f> ^ 2tt\ 



(da . 
= — sn 
J dQ. 



X 



This yields 



pv 



! )X 



(2zZe** 

O ~ 77-1 

\ pv 



1 



J2. 

'mm 



= TTd 



V^min 

JTzZfi 
\ Hv / 



+ 2 ) 2 



(13.103) 



(13.104) 



456 Classical Electrodynamics 

where the final form is obtained by using 0£(j n (13.98). It shows that at 
high velocities the total cross section can be far smaller than the classical 
value of geometrical area ira 2 . 



13.7 Mean Square Angle of Scattering and the Angular Distribution of 
Multiple Scattering 

Rutherford scattering is confined to very small angles even for a point 
Coulomb field, and for fast particles d max is small compared to unity. Thus 
there is a very large probability for small-angle scattering. A particle 
traversing a finite thickness of matter will undergo very many small-angle 
deflections and will generally emerge at a small angle which is the cumu- 
lative statistical superposition of a large number of deflections. Only 
rarely will the particle be deflected through a large angle; since these 
events are infrequent, such a particle will have made only one such 
collision. This circumstance allows us to divide the angular range into 
two regions — one region at comparatively large angles which contains only 
the single scatterings, and one region at very small angles which contains 
the multiple or compound scatterings. The complete distribution in angle 
can be approximated by considering the two regions separately. The 
intermediate region of so-called plural scattering must allow a smooth 
transition from small to large angles. 

The important quantity in the multiple-scattering region, where there 
is a large succession of small-angle deflections symmetrically distributed 
about the incident direction, is the mean square angle for a single scattering. 
This is defined by 

(S 2 ) = J r / Q (13.105) 

J da 

With the approximations of Section 13.6 we obtain 

<0 2 > = 20U In (^) (13.106) 

^ Oram' 

If the quantum value (13.99) of m in is used along with max (13.102), then 
(13.106) has the numerical form: 

<0 2 > ~ 40f nin In (210Z-*) (13.107) 

If nuclear size is unimportant (generally only of interest for electrons, and 
perhaps other particles at very low energies), max should be put equal to 



[Sect. 13.7] 



Collisions between Charged Particles 



457 




Fig. 13.7 
unity in (13.106). Then the argument of the logarithm in (13.107) becomes 



/192 p V* 



instead of (210Z~^). 



It is often desirable to use the projected angle of scattering d', the 
projection being made on some convenient plane such as the plane of a 
photographic emulsion or a bubble chamber, as shown in Fig. 13.7. For 
small angles it is easy to show that 



<0' 2 > = i<0 2 > 



(13.108) 



In each collision the angular deflections obey the Rutherford formula 
(13.92) suitably cut off at min and m ax, with average value zero (when 
viewed relative to the forward direction, or as a projected angle) and mean 
square angle (0 2 ) given by (13.106). Since the successive collisions are 
independent events, the central-limit theorem of statistics can be used to 
show that for a large number n of such collisions the distribution in angle 
will be approximately Gaussian around the forward direction with a mean 
square angle <© 2 ) = n (0 2 ). The number of collisions occurring as the 
particle traverses a thickness / of material containing N atoms per unit 

VOlUme iS tj y 2\2 , 

n = Not~TrN\±^-)-!- (13.109) 

\ BV / 0min 



This means that the mean square angle of the Gaussian is 



<0 2 > ~ 2ttN 



mn 



(13.110) 



P M (d') dd' = -p4== exp (- -||-) dd' (13.112) 



458 Classical Electrodynamics 

Or, using (13.107) for <0 2 >, 

<0 2 > ~ 47tN[^^-) In (2lOZ- 1A )t (13.111) 

\ pv I 

The mean square angle increases linearly with the thickness t. But for 

reasonable thicknesses such that the particle does not lose appreciable 

energy, the Gaussian will still be peaked at very small forward angles. 

The multiple-scattering distribution for the projected angle of scattering 

is 

J. 

V^T"^ <© 2 > 

where both positive and negative values of d' are considered. The small- 
angle Rutherford formula (13.92) can be expressed in terms of the pro- 

JeCt6dangleaS *--(«tf-L (1 ,U3) 

I dd' 2 \ pv / d' 3 

This gives a single-scattering distribution for the projected angle : 

P s (6') dd' = Nt^dd'=^ Nt{ 2 -^^- (13.114) 

SV ' dd' 2 \ pv J d' z 

The single-scattering distribution is valid only for angles large compared 
to (& 2 Y A , and contributes a tail to the Gaussian distribution. 
If we express angles in terms of the relative projected angle, 

a = — ^ (13.115) 

<0 2 )^ V 

the multiple- and single-scattering distributions can be written 



1 _ 2 
P m (ol) da. = —= e a doc 

V 77 " 

_ , N , 1 doc 

P 8 (ol) da = ni ,^^_ v — z 



(13.116) 



81n(210Z-^) a 6 

where (13.1 1 1) has been used for (0 2 >. We note that the relative amounts 
of multiple and single scatterings are independent of thickness in these 
units, and depend only on Z. Even this Z dependence is not marked. The 
factor 8 In (210Z _1 ^) has the value 36.0 for Z = 13 (aluminum) and the 
value 31 .0 for Z = 82 (lead). Figure 13.8 shows the general behavior of the 
scattering distributions as a function of a. The transition from multiple 
to single scattering occurs in the neighborhood of a ^ 2.5. At this point the 
Gaussian has a value of 1/600 times its peak value. Thus the single-scatter- 
ing distribution gives only a very small tail on the multiple-scattering curve. 
There are two things which cause departures from the simple behavior 
shown in Fig. 13.8. The Gaussian shape is the limiting form of the 



[Sect. 13.8] 



Collisions between Charged Particles 



459 




Fig. 13.8 Multiple and single scattering distributions of projected angle. In the region 

of plural scattering (a <-- 2-3) the dotted curve indicates the smooth transition from the 

small-angle multiple scattering (approximately Gaussian in shape) to the wide-angle 

single scattering (proportional to a -3 ). 

angular distribution for very large n. If the thickness / is such that n 
(13.109) is not very large (i.e., n < 100), the distribution follows the single- 
scattering curve to smaller angles than a ^ 2.5, and is somewhat more 
sharply peaked at zero angle than a Gaussian. On the other hand, if the 
thickness is great enough, the mean square angle <0 2 > becomes comparable 
with the angle max (13.102) which limits the angular width of the single- 
scattering distribution. For greater thicknesses the multiple -scattering 
curve extends in angle beyond the single-scattering region, so that there is 
no single-scattering tail on the distribution (see Problem 13.5). 



13.8 Electrical Conductivity of a Plasma 



The considerations of multiple scattering can be applied rather directly 
to the seemingly different problem of the electrical conductivity of a 



460 Classical Electrodynamics 

plasma. For simplicity we will consider the so-called Lorentz gas, which 
consists of TV fixed ions of charge Ze per unit volume and NZ free electrons 
per unit volume. Furthermore electron-electron interactions will be 
ignored. The approximation of fixed ions is a reasonable one, at least for 
plasmas with electrons and ions at roughly the same kinetic temperatures. 
The effects of electron-electron collisions will be mentioned later. 

The simple Drude theory of electrical conductivity, described briefly in 
Section 7.8, is based on the single electron equation, 

dv 
m — = eE — mv\ (13.1 17) 

dt 

where v is the collision frequency. The low-frequency electrical con- 
ductivity a due to electron motion is 

NZe 2 
a = -=^- (13.118) 

mv 

The problem of calculating the proper collision frequency can be ap- 
proached by noting that the term mv\ in (13.1 17) really represents the rate 
of decrease of forward momentum because of Coulomb collisions with the 
ions as the electron moves under the action of the applied electric field. If 
the scattering angle in a single elastic collision is 6, as indicated in Fig. 13.9, 
the forward momentum lost by a particle of momentum/? is p(l — cos 6). 
The average value of this quantity multiplied by the number of collisions 
per unit distance is the loss in forward momentum per unit distance, 
namely, mv. Thus 

mv = Nap (1 - cos d) (13.119) 

where a here is the total cross section (13.104). Since all the Coulomb 
scattering is at very small angles, <1 — cos 6) ~ \ <0 2 ). Then the forward- 
momentum loss per unit distance is 



2\2 



(Ze*> 



mv ~ hNopiQ 2 ) = AttN^- In ^ (13.120) 



mv 



("max I 
B m \J 



Equation (13.106) has been used for (6 2 ). When (13.120) is inserted in 
(13.118), we obtain a conductivity, 

3 

o(v)~ — — (13.121) 

47T(Ze 2 ) In (0 ma x/0min) 

This result holds for electrons of velocity v. 

We now want to average over a thermal distribution. The variation 
with v in (13.121) comes mainly from the factor t' 3 . The argument of the 
logarithm can be evaluated at the mean velocity without introducing 
appreciable error. At energies appropriate to even the hottest plasmas 
nuclear-size effects are negligible. Consequently we put max = 1. The 
value of 0min requires some discussion. For the screened atomic potential 



[Sect. 13.8] Collisions between Charged Particles 461 

P 



Fig. 13.9 

the result (13.97) or (13.98) was appropriate, with the atomic radius a 
given by (13.95). For electron-ion collisions in a plasma the interaction 
is the Debye-Huckel screened potential (10. 113). Consequently the Debye 
length kj)- 1 plays the role of the atomic radius a in the formulas for mln . 
Either (13.97) or (13.98) is used, depending on which is larger. With these 
substitutions the argument of the logarithm in (13.121) can be written 



A = ^ 



12ttJV 



" D (13.122) 

Ze 2 \2itN 



where k D is given by (10.106) or (10.112), and {u 2 Y A = kTjm. The upper 
(lower) value of A is to be used when the mean electron energy f kT is less 
(greater) than 13.6Z 2 electron volts. 

The average value of the «th power of the magnitude of velocity for a 
Maxwellian distribution is , N 

/2JC7T* I 2 ; (m23) 

\ m / T(f ) 

Consequently the value of the conductivity (13.121), averaged over a 
Maxwellian velocity distribution, is 

m (2 kT\ A ..., .... 

<J— — „ (13.124) 

Ze 2 In A W m J 

This approximate result, obtained in a rather simple-minded way with the 
elementary Drude theory, is within a factor of 2 of the correct value found 
from an application of the Boltzmann equation. The physical mechanism 
is the same in both calculations, but the more rigorous treatment involves 
an averaging over v 5 rather than v z * Thus the two results differ by a 
factor (v 5 )l(v 2 )(v 3 ) = 2. 

* The added power of v 2 can be understood as follows. In the presence of the electric 
field the formerly spherically symmetric velocity distribution tends to become distorted 
in velocity directions parallel to the field. The amount of distortion determines the 
current and, through Ohm's law, the conductivity. The distorted distribution results 
from a balancing of the anisotropic electric force and the tendency towards isotropy 
produced by the collisions. Since the scattering cross section varies as v~ 2 , the aniso- 
tropic part of the distribution has more high-velocity components than normal by a 
factor y 2 . 



462 Classical Electrodynamics 

When electron-electron collisions are included, the forward-momentum 
loss is increased and so the conductivity is decreased from its value of 
twice (13.124). The relative decrease depends on Z roughly as Z/(l + Z), 
ranging from 0.58 for Z = 1 to 1.0 for Z-> oo. Consequently (13.124) 
as it stands can be used as a good approximation to the conductivity for a 
hydrogen or deuterium plasma, including effects of electron-electron 
collisions. If the classical (low-energy) value of A (13.122) is used, (13.124) 
can be written in the instructive form : 



\(-)"A-<», 3125 ) 

3 \77/ 



3 W/ In A 

Since A is of the order of 10 4 for a typical hydrogen plasma (n e ~ 10 15 
cm- 3 , T~ 10 5 °K), a is ^200^ ^4x 10 14 sec" 1 . This is not quite as 
large as metallic conductivities (~10 16 sec -1 ), but is sufficiently large that 
the infinite conductivity approximation used in Chapter 10 is quite 
adequate in discussing the penetration of fields into a plasma. 



REFERENCES AND SUGGESTED READING 

The problems of the penetration of particles through matter have interested Niels 
Bohr all his life. A lovely presentation of the whole subject, with characteristic emphasis 
on the interplay of classical and quantum-mechanical effects, appears in his compre- 
hensive review article of 1948: 

Bohr. 
Numerical tables and graphs of energy-loss data, as well as key formulas, are given by 
Rossi, Chapter 2, 

Segre, article by H. A. Bethe and J. Ashkin. 
Rossi also gives a semiclassical treatment of energy loss and scattering similar to ours. 
The density effect on the energy loss by extremely relativistic particles is discussed, 
with numerous results for different substances in graphical form, by 
R. M. Sternheimer, Phys. Rev., 88, 851 (1952); 91, 256 (1953). 
The correct calculation of the conductivity of a plasma is outlined by 
Spitzer, Chapter 5. 



PROBLEMS 

13.1 A heavy particle of charge ze, mass M, and nonrelativistic velocity v 
collides with a free electron of charge e and mass m initially at rest. With 
no approximations, other than that of nonrelativistic motion and M > m, 
show that the energy transferred to the electron in this Coulomb collision, 
as a function of the impact parameter b, is 

2(ze 2 ) 2 1 



nw 2 b 2 + {ze 2 jmv 2 ) 2 



[Probs. 13] Collisions between Charged Particles 463 

13.2 {a) Taking/j<o>> = 12Zevin the quantum-mechanical energy-loss formula, 
calculate the rate of energy loss (in Mev/cm) in air at NTP, aluminum, 
copper, lead for a proton and a mu meson, each with kinetic energies of 
10, 100, 1000 Mev. 

(b) Convert your results to energy loss in units of Mev-cm 2 /gm and 
compare the values obtained in different materials. Explain why all the 
energy losses in Mev-cm 2 /gm are within a factor of 2 of each other, whereas 
the values in Mev/cm differ greatly. 

13.3 Consider the energy loss by close collisions of a fast, but nonrelativistic, 
heavy particle of charge ze passing through an electronic plasma. Assume 
that the screened Coulomb interaction (10.113) acts between the electrons 
and the incident particle. 

(a) Show that the energy transfer in a collision at impact parameter b is 
given approximately by 

AE(b) ^^^ k^K^knb) 

where m is the electron mass, v is the velocity of the incident particle, and 
Icd is the Debye wave number (10.112). 

(b) Determine the energy loss per unit distance traveled for collisions 
with impact parameter greater than b min . Assuming kDb min < 1, write 
down your result with both the classical and quantum-mechanical values 
of b min . 

13.4 With the same approximations as were used to discuss multiple scattering, 
show that the projected transverse displacement y (see Fig. 13.7) of an 
incident particle is described approximately by a Gaussian distribution, 



P(y) dy = A exp 



-y* 

2<2/ 2 >. 



dy 



where the mean square displacement is (y 2 ) = (x 2 /6)<0 2 >, x being the 
thickness of the material traversed and <© 2 > the mean square angle of 
scattering. 
13.5 If the finite size of the nucleus is taken into account in the "single-scattering" 
tail of the multiple-scattering distribution, there is a critical thickness x c 
beyond which the single-scattering tail is absent. 

(a) Define x e in a reasonable way and calculate its value (in cm) for 
aluminum and lead, assuming that the incident particle is relativistic. 

(b) For these thicknesses calculate the number of collisions which occur 
and determine whether the Gaussian approximation is valid. 



14 



Radiation by Moving Charges 



It is well known that accelerated charges emit electromagnetic 
radiation. In Chapter 9 we discussed examples of radiation by macroscopic 
time-varying charge and current densities, which are fundamentally charges 
in motion. We will return to such problems in Chapter 16 where multipole 
radiation is treated in a systematic way. But there is a class of radiation 
phenomena where the source is a moving point charge or a small number 
of such charges. In these problems it is useful to develop the formalism in 
such a way that the radiation intensity and polarization are related 
directly to properties of the charge's trajectory and motion. Of particular 
interest are the total radiation emitted, the angular distribution of radiation, 
and its frequency spectrum. For nonrelativistic motion the radiation is 
described by the well-known Larmor result (see Section 14.2). But for 
relativistic particles a number of unusual and interesting effects appear. 
It is these relativistic aspects which we wish to emphasize. In the present 
chapter a number of general results are derived and applied to examples of 
charges undergoing prescribed motions, especially in external force fields. 
Chapter 15 deals with radiation emitted in atomic or nuclear collisions. 



14.1 Lienard-Wiechert Potentials and Fields for a Point Charge 

In Chapter 6 it was shown that for localized charge and current distri- 
butions without boundary surfaces the scalar and vector potentials can be 
written as 

AJx, t) = - j\ Jtl(X ' ,t) d ( t ' + ~ - ') #*' dt' (14.1) 

where R = (x — x'), and the delta function provides the retarded behavior 

464 



[Sect. 14.1] Radiation by Moving Charges 465 

demanded by causality. For a point charge e with velocity c(S(?) at the 
point r(7) the charge-current density is 

J Ax, = ec^ d[x - r(0] (14.2) 

where ^ = (p, /)• With tnis source density the spatial integration in (14.1) 
can be done immediately, yielding 



A ax, t) = e\ — — o 



f' + 



R(t') 



— t 



dt' 



(14.3) 



where now R(t') = \x — r(t')\. Although (14.3) is a convenient form to 
utilize in calculating the fields, the integral over dt' can be performed, 
provided we recall from Section 1.2 that when the argument of the delta 
function is a function of the variable of integration the standard results are 
modified as follows : 



/■ 



g(*) d[f( x ) -ci]dx = 



g( x ) 



UfldxJ 
The function f(t') = t' + [R(t')/c] has a derivative 



f(x) = a 



df 1 , ldR , a 



(14.4) 



(14.5) 



dt' c dt' 

where cp is the instantaneous velocity of the particle, and n = R/i? is a 
unit vector directed from the position of the charge to the observation 
point. With (14.5) in (14.4) and (14.3) the potentials of the point charge, 
called the Lienard- Wiechert potentials, are 



0(x, t) = e 



A(x, t) = e 



J_ 

A 

kR. 



(14.6) 



The square bracket with subscript ret means that the quantity in brackets 
is to be evaluated at the retarded time, t' = t — [R(t)/c]. We note that, 
for nonrelativistic motion, k -» 1. Then the potentials (14.6) reduce to the 
well-known nonrelativistic results. 

To determine the fields E and B from the potentials A^ it is possible to 
perform the specified differential operations directly on (14.6). But this is 
a more tedious procedure than working with the form (14.3). We note that 
in (14.3) the only dependence on the spatial coordinates x of the observa- 
tion point is through R. Hence the gradient operation is equivalent to 

d d 



MR 



dR 



= n 



dR 



(14.7) 



466 Classical Electrodynamics 

Consequently the electric and magnetic fields can be written as 

d 



dt' 



(14.8) 



f X + ^-f) 

B(x,r) = . (nx(3) -— £ - + J-«5'(V+*-A 

The primes on the delta functions mean differentiation with respect to 
their arguments. If the variable of integration is changed to/(r') = t' + 
[R(t')/c], we can integrate by parts on the derivative of the delta function. 
Then we find readily 



E(x, = e 
B(x, t) = e 



+ J_ d_ (n - p 



ret 



.kR 2 ck dt' \ kR 



> x n [ 1 d / ft x n \" 

. kR 2 CKdt' \ kR /_ 



ret 



(14.9) 



It is convenient to perform first the differentiation of the unit vector n. 
It is evident from Fig. 14.1 that the rate of change of n with time is the 
negative of the ratio of the perpendicular component of v to R. Thus 



_rfn _ n x (n x P) 

cdt' R 



(14.10) 



When we perform the differentiation of n wherever it appears explicitly, 
we obtain 



E(x, t) = e 
B(x, = e 



"_E_ + H_± l±\ _ JL _ 1 A (IX 

-K 2 R* CKdt'\KRJ K 2 R 2 CKdt'\KR!. 

\k 2 R 2 CKdt'\KR/J 



(14.11) 



We observe at this point that the magnetic induction is related simply to 



Mt') 




Fig. 14.1 



[Sect. 14.1] 



Radiation by Moving Charges 



467 



the electric field by the relation, 

B = n x E (14.12) 

where the equation is understood to be in terms of the retarded quantities 
in square brackets. 

The remaining derivatives needed in (14.11) are 



dt' 
cdt' c 



Then the electric field can be written 



E(x, = e 



(n - pXl-i 2 )" 



ret c 



-f- x {(n - 0) x p} 
Lk 3 R 



_ ret 



(14.13) 



(14.14) 



while the magnetic induction is given by (14.12). Fields (14.12) and (14.14) 
divide themselves naturally into "velocity fields," which are independent 
of acceleration, and "acceleration fields," which depend linearly on p. 
The velocity fields are essentially static fields falling off as Rr 2 , whereas 
the acceleration fields are typical radiation fields, both E and B being 
transverse to the radius vector and varying as U -1 . 

For a particle in uniform motion the velocity fields must be the same as 
those obtained in Section 11.10 by means of a Lorentz transformation on 
the static Coulomb field. For example, the transverse electric field E x at a 
point a perpendicular distance b from the straight line path of the charge 
was found to be 

EJt) = ey o3 , (14.15) 

The origin of the time t is chosen so that the charge is closest to the 
observation point at t = 0. The electric field E t (t) given by (14.15) bears 
little resemblance to the velocity field in (14.14). The reason for this 
apparent difference is that field (14.15) is expressed in terms of the present 
position of the charge rather than its retarded position. To show the 
equivalence of the two expressions we consider the geometrical configura- 
tion shown in Fig. 14.2. Here O is the observation point, and the points P 
and P' are the present and apparent or retarded positions of the charge at 
time t. The distance P'Q is pR cos = (n • p)fl. Therefore the distance 
OQ is kR. But from triangles OPQ and PP'Q we find 

( K Rf = r 2 - (PQ) 2 = r 2 - p*(R sin 0) 2 
Then from triangle OMP' we have R sin 6 = b, so that 

(kR) 2 =b 2 + vH 2 - A 2 = " 2 (b 2 + A 2 * 2 ) (14.16) 



468 



Classical Electrodynamics 




Fig. 14.2 Present and retarded positions of a charge in uniform motion. 



The transverse component of the velocity field in (14.14) is 

b 



£i(0 = e 



.y\ K RfS 



ret 



(14.17) 



With substitution (14.16) for kR in terms of the charge's present position, 
we find that (14.17) is equal to (14.15). The other components of E and B 
come out similarly. 



14.2 Total Power Radiated by an Accelerated Charge — Larmor's 
Formula and Its Relativistic Generalization 

If a charge is accelerated but is observed in a reference frame where its 
velocity is small compared to that of light, then in that coordinate frame 
the acceleration field in (14.14) reduces to 

n x (n x (J) 



E„ = e - 



C L 



R 



The instantaneous energy flux is given by the Poynting's vector, 
S = -ExB=- |EJ 2 n 

47T 4"7T 

This means that the power radiated per unit solid angle is* 

§■ = y ItfEJ 2 =/-i«x(nx p)| 2 
ail 47r 4nc 

* In writing angular distributions of radiation we will always exhibit the polariza- 
tion explicitly by writing the absolute square of a vector which is proportional to the 
electric field. 



(14.18) 



(14.19) 



(14.20) 



[Sect. 14.2] 



Radiation by Moving Charges 



469 



If is the angle between the acceleration v and n, as shown in Fig. 14.3, 
then the power radiated can be written 



dP 
dQ. 



= J—i? sin 2 

47TC 3 



(14.21) 



This exhibits the characteristic sin 2 angular dependence which is a well- 
known result. We note from (14.18) that the radiation is polarized in the 
plane containing v and n. The total instantaneous power radiated is 
obtained by integrating (14.21) over all solid angle. Thus 



3 c 3 



(14.22) 



This is the familiar Larmor result for a nonrelativistic, accelerated charge. 
Larmor's formula (14.22) can be generalized by arguments about 
covariance under Lorentz transformations to yield a result which is valid 
for arbitrary velocities of the charge. Radiated electromagnetic energy 
behaves under Lorentz transformation like the fourth component of a 
4-vector (see Problem 11.13). Since dE mA = P dt, this means that the 
power P is a Lorentz invariant quantity. If we can find a Lorentz invariant 
which reduces to the Larmor formula (14.22) for £ < 1, then we have the 
desired generalization. There are, of course, many Lorentz invariants 
which reduce to the desired form when £->-0. But from (14.14) it is 
evident that the general result must involve only p and p. With this 
restriction on the order of derivatives which can appear the result is unique. 
To find the appropriate generalization we write Larmor's formula in the 
suggestive form : 

" 2 f^E.^) (14.23) 

\dt dt! 



P = 2 - 




Fig. 14.3 



470 Classical Electrodynamics 

where m is the mass of the charge, and p its momentum. The Lorentz 
invariant generalization is clearly 

P = ? 4- 3 l^) (14.24) 

3 ra V V dr dr! 

where dr = dtjy is the proper time element, and/?^ is the charged particle's 
momentum-energy 4- vector.* To check that (14.24) reduces properly to 
(14.23) as /? -■> we evaluate the 4-vector scalar product, 

dp,dp IL= (dpf_i (dE) 2 = (dpY _ * (dpY (u 2$) 

dr dr \drf c 2 \dr I W H \dr! 

If (14.24) is expressed in terms of the velocity and acceleration by means 
of E = ymc 2 and p = ytm, we obtain the Lienard result (1898): 

i } = ;-7W-(P><M (14.26) 

3 c 

One area of application of the relativistic expression for radiated power 
is that of charged-particle accelerators. Radiation losses are sometimes 
the limiting factor in the maximum practical energy attainable. For a 
given applied force (i.e., a given rate of change of momentum) the radiated 
power (14.24) depends inversely on the square of the mass of the particle 
involved. Consequently these radiative effects are largest for electrons. 
We will restrict our discussion to them. 

In a linear accelerator the motion is one dimensional. From (14.25) it 
is evident that in that case the radiated power is 

P = \4- iff (14.27) 

3 m 2 c 3 \dt I 

The rate of change of momentum is equal to the change in energy of the 
particle per unit distance. Consequently 

P== ?jM^) 2 (14.28) 

3 mV \dx) K 

showing that for linear motion the power radiated depends only on the 
external forces which determine the rate of change of particle energy with 
distance, not on the actual energy or momentum of the particle. The ratio 

* That (14.24) is unique can be seen by noting that a Lorentz invariant is formed by 
taking scalar products of 4-vectors or higher-rank tensors. The available 4-vectors are 
Pp and dp^/ch. Only form (14.24) reduces to the Larmor formula for /S -»■ 0. Contraction 
of higher-rank tensors such as p^idpjdr) can be shown to vanish, or to give results 
proportional to (14.24) or m 2 . 



[Sect. 14.2] Radiation by Moving Charges 471 

of power radiated to power supplied by the external sources is 

__P_ = 2 J_ 1 dE _ 2 (e 2 lmc 2 ) dE (J4 29) 

(d£/dO ~~ 3 m 2 c 3 i> d* 3 mc 2 dx 

where the last form holds for relativistic particles (/5->l). Equation (14.29) 
shows that the radiation loss will be unimportant unless the gain in 
energy is of the order of mc 2 = 0.511 Mev in a distance of e>c 2 = 
2.82 x 10- 13 cm, or of the order of 2 X 10 14 Mev/meter! Typical energy 
gains are less than 10 Mev/meter. Radiation losses are completely 
negligible in linear accelerators. 

Circumstances change drastically in circular accelerators like the 
synchrotron or betatron. In such machines the momentum p changes 
rapidly in direction as the particle rotates, but the change in energy per 
revolution is small. This means that 



dp 

dr 



= yco\v\>-^ (14-30) 

c dr 



Then the radiated power (14.24) can be written approximately 

p=?-4i^ 2 ipi 2 =? £ T^ 4 (1431) 

3 mV 3 p* 

where we have used co = (cft/p), p being the orbit radius. This result was 
first obtained by Lienard in 1898. The radiative-energy loss per revolution 
is 

dE = llL£p = ±L e lpy (14.32) 

cP 3 P 

For high-energy electrons (/? ^ 1) this has the numerical value, 

dE (Mev) = 8.85 X 10" 2 [£(Bev)] ' (14.33) 

p (meters) 

For a typical low-energy synchroton, p ^ 1 meter, .E max ^0.3 Bev. 
Hence, (5£ max ~ 1 Kev per revolution. This is less than, but not negligible 
compared to, the energy gain of a few kilovolts per turn. In the largest 
electron synchrotrons, the orbit radius is of the order of 10 meters and the 
maximum energy is 5 Bev. Then the radiative loss is ~5.5 Mev per revolu- 
tion. Since it is extremely difficult to generate radiofrequency power at 
levels high enough to produce energy increments much greater than this 
amount per revolution, it appears that 5-10 Bev is an upper limit on the 
maximum energy of circular electron accelerators. 



472 Classical Electrodynamics 

The power radiated in circular accelerators can be expressed numerically 
as 

_, „. 10 6 <5£(Mev) _, . ,<aia\ 

P (watts) = J (amp) (14.34) 

2tt p (meters) 

where / is the circulating beam current. This equation is valid if the 
emission of radiation from the different electrons in the circulating beam 
is incoherent. In the largest electron synchrotrons the radiated power 
amounts to 0.1 watt per microampere of beam. Although this power 
dissipation is very small the radiated energy is readily detected and has 
some interesting properties which will be discussed in Section 14.6. 



14.3 Angular Distribution of Radiation Emitted by an 
Accelerated Charge 

For an accelerated charge in nonrelativistic motion the angular distri- 
bution shows a simple sin 2 behavior, as given by (14.21), where is 
measured relative to the direction of acceleration. For relativistic motion 
the acceleration fields depend on the velocity as well as the acceleration. 
Consequently the angular distribution is more complicated. From (14.14) 
the radial component of Poynting's vector can be calculated to be 



e 2 

[S • n]ret = 

Aire 



1 n x [(n - p) x p]| 



U«R 2 



ret 



(14.35) 



It is evident that there are two types of relativistic effect present. One is 
the effect of the specific spatial relationship between (3 and (3, which will 
determine the detailed angular distribution. The other is a general, relati- 
vistic effect arising from the transformation from the rest frame of the 
particle to the observer's frame and manifesting itself by the presence of 
the factors k (14.5) in the denominator of (14.35). For ultrarelativistic 
particles the latter effect dominates the whole angular distribution. 

In (14.35) S • n is the energy per unit area per unit time detected at an 
observation point at time t due to radiation emitted by the charge at time 
t' = t — R{t')jc, If we wanted to calculate the energy radiated during a 
finite period of acceleration, say from /' = T x to t' = T 2 , we would write 

W=\ [S-n] ret ^= (S-n) ^dt' (14.36) 

Thus we see that the useful and meaningful quantity is (S • n) (dt/dt'), the 
power radiated per unit area in terms of the charge's own time. We 



[Sect. 14.3] 



Radiation by Moving Charges 



473 



Fig. 14.4 Radiation pattern for 
charge accelerated in its direction 
of motion. The two patterns are 
not to scale, the relativistic one 
(appropriate for y — 2) having 
been reduced by a factor ~10 2 
for the same acceleration. 



therefore define the power radiated per unit solid angle to be 




dP(n = R\S.n)— = KR 2 S'n 



dQ. 



dt 
dt' 



(14.37) 



If we imagine the charge to be accelerated only for a short time during 
which |3 and (3 are essentially constant in direction and magnitude, and we 
observe the radiation far enough away from the charge that n and R 
change negligibly during the acceleration interval, then (14.37) is pro- 
portional to the angular distribution of the energy radiated. With (14.35) 
for the Poynting's vector, the angular distribution is 



dP(t') _ e* In x {(n - ft) x (3}| 2 



da 



Attc (1 - n • p) 5 



(14.38) 



The simplest example of (14.38) is linear motion in which (3 and (3 are 
parallel. If 6 is the angle of observation measured from the common 
direction of (3 and (3, then (14.38) reduces to 



dP(t') 
dQ 



e v 



sin 



4ttc 3 (1 - cos Of 



(14.39) 



For /? < 1, this is the Larmor result (14.21). But as /S -> 1, the angular 
distribution is tipped forward more and more and increases in magnitude, 
as indicated schematically in Fig. 14.4. The angle max for which the 
intensity is a maximum is 



= cos 



J-(Vl + 15/P-l) 

L3p 



2y 



(14.40) 



where the last form is the limiting value for /S — >• 1 . In this same limit the 
peak intensity is proportional to y 8 . Even for /3 = 0.5, corresponding to 
electrons of <~80 Kev kinetic energy, m ax = 38.2°. For relativistic 
particles, m ax is very small, being of the order of the ratio of the rest 
energy of the particle to its total energy. Thus the angular distribution is 



474 



Classical Electrodynamics 



confined to a very narrow cone in the direction of motion. For such small 
angles the angular distribution (14.39) can be written approximately 



dQ. 77 



(ydf 



(i + y 2 ey 



(14.41) 



The natural angular unit is evidently y _1 . The angular distribution is 
shown in Fig. 14.5 with angles measured in these units. The peak occurs 
at yd = |, and the half-power points at yd = 0.23 and yd = 0.91. The 
root mean square angle of emission of radiation in the relativistic limit is 



(d 2 ) 1A = A = — 
y E 



(14.42) 



This is typical of the relativistic radiation patterns, regardless of the 
vectorial relation between (3 and (3. The total power radiated can be 
obtained by integrating (14.39) over all angles. Thus 



2 p 2 
P(t') = -"-vY 

3 c 



(14.43) 



in agreement with (14.26) and (14.27). 

Another example of angular distribution of radiation is that for a charge 
in instantaneously circular motion with its acceleration (3 perpendicular 
to its velocity (3. We choose a coordinate system such that instantaneously 
(3 is in the z direction and (3 is in the x direction. With the customary polar 
angles 6, <f> defining the direction of observation, as shown in Fig. 14.6, the 
general formula (14.38) reduces to 



dP(t') 
dQ. 



1 



4ttc 3 (1-/9 cos Of 



1 - 



sin 2 6 cos 2 <£ 



, 2 (1 



P cos Of A 



(14.44) 



We note that, although the detailed angular distribution is different from 
the linear acceleration case, the same characteristic relativistic peaking at 
forward angles is present. In the relativistic limit (y > 1), the angular 




Fig. 14.5 Angular distribution of 
radiation for relativistic particle. 



[Sect. 14.4] 



Radiation by Moving Charges 



475 




distribution can be written approximately 



dP(t') 



l e JL 

77 C 3 



1 



1 - 



4y 2 2 cos 2 <f 

(1 + r 202 )2 j 



(14.45) 



dQ. 77 c 3 ' (1 + y 2 2 ) 3 
The root mean square angle of emission in this approximation is given by 
(14.42), just as for one-dimensional motion. The total power radiated can 
be found by integrating (14.44) over all angles or from (14.26): 



«0 = lfr> 



(14.46) 



It is instructive to compare the power radiated for acceleration parallel 
to the velocity (14.43) or (14.27) with the power radiated for acceleration 
perpendicular to the velocity (14.46) for the same magnitude of applied 
force. For circular motion, the magnitude of the rate of change of 
momentum (which is equal to the applied force) is ymv. Consequently, 
(14.46) can be written 2 



■» circular \* ) — 



3 mV 



r 



©• 



(14.47) 



When this is compared to the corresponding result (14.27) for rectilinear 
motion, we find that for a given magnitude of applied force the radiation 
emitted with a transverse acceleration is a factor of y % larger than with a 
parallel acceleration. 



14.4 Radiation Emitted by a Charge in Arbitrary, Extreme Relativistic 
Motion 

For a charged particle undergoing arbitrary, extreme relativistic motion 
the radiation emitted at any instant can be thought of as a coherent super- 
position of contributions coming from the components of acceleration 



476 Classical Electrodynamics 

parallel to and perpendicular to the velocity. But we have just seen that for 
comparable parallel and perpendicular forces the radiation from the parallel 
component is negligible (of order 1/y 2 ) compared to that from the perpen- 
dicular component. Consequently we may neglect the parallel component 
of acceleration and approximate the radiation intensity by that due to the 
perpendicular component alone. In other words, the radiation emitted by 
a charged particle in arbitrary, extreme relativistic motion is approxi- 
mately the same as that emitted by a particle moving instantaneously along 
the arc of a circular path whose radius of curvature p is given by 

v 2 c 2 
P = — ~— (14.48) 

where v x is the perpendicular component of acceleration. The form of the 
angular distribution of radiation is (14.44) or (14.45). It corresponds to a 
narrow cone or searchlight beam of radiation directed along the instanta- 
neous velocity vector of the charge. 

For an observer with a frequency-sensitive detector the confinement of 
the radiation to a narrow pencil parallel to the velocity has important 
consequences. The radiation will be visible only when the particle's 
velocity is directed towards the observer. For a particle in arbitrary 
motion the observer will detect a pulse or burst of radiation of very short 
time duration (or a succession of such bursts if the particle is in periodic 
motion), as sketched in Fig. 14.7. Since the angular width of the beam is of 
the order of y 1 , the particle will illuminate the observer only for a time 
interval 

cy 
in terms of its own time, where p is the radius of curvature (14.48). The 
observer sees, however, a time interval, 

\dt'/ 

where (dt/dt') = (k) ~ (1/y 2 ). Consequently the duration of the burst of 

radiation at the detector is 

Af ~ -i £ (14.49) 

y 6 c 

A pulse of this duration will contain, according to general arguments about 
Fourier integrals (see Section 7.3), appreciable frequency components up to 
a critical frequency, co c , of the order of 



^~(-)y 3 04.50) 



For circular motion cjp is the angular frequency of rotation co , and even 



[Sect. 14.5] 



Radiation by Moving Charges 



All 




— — — O 



Pit) 




Koo) 




Fig. 14.7 Radiating particle illuminates the detector at O only for a time A/. The 
frequency spectrum thus contains frequencies up to a maximum io c ~ (Af) _1 . 

for arbitrary motion it plays the role of a fundamental frequency of motion. 
Equation (14.50) shows that a relativistic particle emits a broad spectrum 
of frequencies if E > mc 2 , up to y 3 times the fundamental frequency. In 
a 200-Mev synchrotron, y m ax ^ 400. Therefore co c ~ 6 x 10 7 eo . Since 
the rotation frequency is a> ~ 3 X 10 8 sec -1 , the frequency spectrum 
of emitted radiation extends up to ~2 x 10 16 sec -1 . This represents a 
wavelength of 1000 angstroms. Hence the spectrum extends beyond the 
visible, even though the fundamental frequency is in the 100-Mc range. 
In Section 14.6 we will discuss in detail the angular distribution of the 
different frequency components, as well as the total energy radiated as a 
function of frequency. 



14.5 Distribution in Frequency and Angle of Energy Radiated by 
Accelerated Charges 



The qualitative arguments of the previous section show that for relati- 
vistic motion the radiated energy is spread over a wide range of frequencies. 



478 Classical Electrodynamics 

The range of the frequency spectrum was estimated by appealing to 
properties of Fourier integrals. The argument can be made precise and 
quantitative by the use of Parseval's theorem of Fourier analysis. 
The general form of the power radiated per unit solid angle is 

^=|A«P (.4.51) 

where 

A(0 = (^-j 2 [KE] re t (14.52) 

E being the electric field (14.14). In (14.51) the instantaneous power is 
expressed in the observer's time (contrary to the definition in Section 14.3), 
since we wish to consider a frequency spectrum in terms of the observer's 
frequencies. For definiteness we think of the acceleration occurring for 
some finite interval of time, or at least falling off for remote past and 
future times, so that the total energy radiated is finite. Furthermore, the 
observation point is considered far enough away from the charge that the 
spatial region spanned by the charge while accelerated subtends a small 
solid angle element at the observation point. 

The total energy radiated per unit solid angle is the time integral of 
(14.51): 

^=J_JA(0l 2 ^ (14.53) 

This can be expressed alternatively as an integral over a frequency 
spectrum by use of Fourier transforms. We introduce the Fourier transform 
A(co) of A(t), 



A(eo) = -L| A(t)e i0it dt (14.54) 

•v Z7T J -co 



and its inverse, 



Then (14.53) can be written 



A(0 = -L J A(w)e~ ilot dm (14.55) 

a Z7T J -oo 



dw i r°° c° f°° 

— - = — dt\ dco] dco' A*(o>') • A{(o)e i{( °'- m)t (14.56) 

dil Z7T J -co J -oo J -co 

Interchanging the orders of time and frequency integration, we see that the 
time integral is just a Fourier representation of the delta function 
d((o' — o>). Consequently the energy radiated per unit solid angle becomes 

4? = I \Mco)\ 2 dco (14.57) 

d\l J -co 



dQ Jo 



[Sect. 14.5] Radiation by Moving Charges 479 

The equality of (14.57) and (14.53), with suitable mathematical restrictions 
on the function A(0, is a special case of Parseval's theorem. It is customary 
to integrate only over positive frequencies, since the sign of the frequency 
has no physical meaning. Then the relation, 

d -^da> (14.58) 

dQ. 

defines a quantity dI{(o)jdQ which is the energy radiated per unit solid 
angle per unit frequency interval : 

^ = |A(co)| 2 + |A(-co)| 2 (14.59) 

dQ. 

If A(0 is real, from (14.55) it is evident that A(-co) = A*(o>). Then 

^^ = 2 |A(o>)| 2 (14.60) 

dQ 

This result relates in a quantitative way the behavior of the power radiated 
as a function of time to the frequency spectrum of the energy radiated. 
By using (14.14) for the electric field of an accelerated charge we can 
obtain a general expression for the energy radiated per unit solid angle 
per unit frequency interval in terms of an integral over the trajectory of 
the particle. We must calculate the Fourier transform (14.54) of A(0 given 
by (14.52). Using (14.14), we find 

A^ ( * V f W ,;4 nx[(n-(3)x $]" 

where ret means evaluated at t' + [R(t')/c] = t. We change the variable 
of integration from t to t', thereby obtaining the result: 

Ma) = (4.)* r - ^ + i«™ ■»[(— p*M „, (14 . 62) 

\87rV J-oc KT 

Since the observation point is assumed to be far away from the region of 
space where the acceleration occurs, the unit vector n is sensibly constant 
in time. Furthermore the distance R(t') can be approximated as 

R(t') ~ x - n • r(0 (14.63) 

where x is the distance from an origin O to the observation point P, and 
r(r') is the position of the particle relative to O, as shown in Fig. 14.8. Then, 
apart from an overall phase factor, (14.62) becomes 

A((o) = (AT* f 00 ^i-[..ko/.]) nx[(n-P)xM df (14 64) 

WV J -00 K 2 

The primes on the time variable have been omitted for brevity. The 



dt (14.61) 

_ ret 



480 



Classical Electrodynamics 
R(t') 




Fig. 14.8 

energy radiated per unit solid angle per unit frequency interval (14.60) is 
accordingly 



dl(a>) 
d£l 



4tt 2 c 



f * n x [(n - (3) x 

J-oo (1-S-iO 2 



P] i«>(t-[n-t(t)lc]) 



dt 



(14.65) 



For a specified motion r(t) is known, (3(?) and (3(0 can be computed, and the 
integral can be evaluated as a function of co and the direction of n. If 
accelerated motion of more than one charge is involved, a coherent sum 
of amplitudes A/co), one for each charge, must replace the single amplitude 
in (14.65) (see Problems 14.11, 15.2, and 15.3). 

Even though (14.65) has the virtue that the time interval of integration 
is explicitly shown to be confined to times for which the acceleration is 
different from zero, a simpler expression for some purposes can be 
obtained by an integration by parts in (14.64). It is easy to demonstrate 
that the integrand in (14.64), excluding the exponential, is a perfect 
differential : 



(14.66) 



n x [(n - 

K 2 


P) x M _ d 
dt 


n x (n x P) 

K 




ration by parts leads to the intensity distribut 


dI(co) _ e 2 eo 2 
dtl 4tt 2 c 


n x (n x 

J — oo 


q\ i(o(t-[n-t(t)lc 


]) dt 



(14.67) 



It should be observed that in (14.67) and (14.65) the polarization of the 
emitted radiation is specified by the direction of the vector integral in each. 
The intensity of radiation of a certain fixed polarization can be obtained 
by taking the scalar product of the appropriate unit polarization vector 
with the vector integral before forming the absolute square. 

For a number of charges e i in accelerated motion the integrand in (14.67) 
involves the replacement, 

*\e$fi- m ' l *' m (14.68) 

3=1 



efie 



—i(co/c)n-t(t) 



[Sect. 14.6] Radiation by Moving Charges 481 

In the limit of a continuous distribution of charge in motion the sum over 
j becomes an integral over the current density J(x, t) : 

e p e -iMOn-m _^ 1 t d s x J(x? f)e -.•(«/«)«•« (14#69) 

Then the intensity distribution becomes 



^ = -^|LUnx[nxJ(x > 0> 
d£l 4wV \J J 



ia>[t-(n-x)lc] 



(14.70) 



a result which can be obtained from the direct solution of the inhomo- 
geneous wave equation for the vector potential (14.1). 

Of some interest is the radiation associated with a moving magnetic 
moment. This can be most easily expressed by recalling from Chapter 5 
that a magnetization density M(x., t) is equivalent to a current, 

J M = C V x M (14.71) 

Then substitution into (14.70) yields 



dI M (co)_ a, 4 \j dt j^ xnxJ(iXtt)e ^-in^ 



d£l 4tt 2 c 



2„3 



(14.72) 



If the magnetization is a point magnetic moment (*,(?) at the point r(t), 
th6n "(x, = tfO d[x - 1(01 (14-73) 



and the energy radiated per unit solid angle per unit frequency interval is 



f M («>) _ ft) 4 I f 
dO. ~4tt 2 c 3 J 



Anx^- 1 '*' 



(14.74) 



dl, 

dO. 

We note that there is a characteristic difference of a factor co 2 between the 
radiated intensity from a magnetic dipole and an accelerated charge, apart 
from the frequency dependence of the integrals. 

The general formulas developed in this section, especially (14.65) and 
(14.67), will be applied in this chapter and subsequent ones to various 
problems involving the emission of radiation. The magnetic-moment 
formula (14.74) will be applied to the problem of radiation emitted in 
orbital-electron capture by nuclei in Chapter 15. 



14.6 Frequency Spectrum of Radiation Emitted by a Relativistic Charged 
Particle in Instantaneously Circular Motion 

In Section 14.4 we saw that the radiation emitted by an extremely 
relativistic particle subject to arbitrary accelerations is equivalent to that 



482 



Classical Electrodynamics 




Fig. 14.9 



emitted by a particle moving instantaneously at constant speed on an 
appropriate circular path. The radiation is beamed in a narrow cone in 
the direction of the velocity vector, and is seen by the observer as a short 
pulse of radiation as the searchlight beam sweeps across the observation 
point. 

To find the distribution of energy in frequency and angle it is necessary 
to calculate the integral in (14.67). Because the duration of the pulse 
A?' ~ (p/cy) is very short, it is necessary to know the velocity (3 and 
position r(t) over only a small arc of the trajectory whose tangent points 
in the general direction of the observation point. Figure 14.9 shows an 
appropriate coordinate system. The segment of trajectory lies in the x-y 
plane with instantaneous radius of curvature p. Since an integral will be 
taken over the path, the unit vector n can be chosen without loss of 
generality to lie in the x-z plane, making an angle 6 (the colatitude) with 
the x axis. Only for very small 6 will there be appreciable radiation 
intensity. The origin of time is chosen so that at t = the particle is at the 
origin of coordinates. 

The vector part of the integrand in (14.67) can be written 



n x (n x p) = £ 



-€,, sin 



(-) + e± cos (-) sin d (14.75) 



where e„ = e 2 is a unit vector in the y direction, corresponding to polariza- 
tion in the plane of the orbit; c ± = n x e 2 is the orthogonal polariza- 
tion vector corresponding approximately to polarization perpendicular to 
the orbit plane (for 6 small). The argument of the exponential is 



CO 



(-■-f>)-[«-H 2 ) 



cos 



(14.76) 



483 



[Sect. 14.6] Radiation by Moving Charges 

Since we are concerned with small angles and comparatively short times 
around t = 0, we can expand both trigonometric functions in (14.76) to 
obtain 



{-^Mfa ')'+&< 



3 



(14.77) 



where (3 has been put equal to unity wherever possible. Using the time 
estimate p\cy for / and the estimate (d 2 Y A (14.42) for 0, it is easy to see that 
neglected terms in (14.77) are of the order of y~ 2 times those kept. 

With the same type of approximations in (14.75) as led to (14.77), the 
radiated-energy distribution (14.67) can be written 



dI(co) 



dO. 4n 2 c 
where the amplitudes are* 



-e H A H (m) + € 1 i4 1 (ft)) 



(14.78) 



^»>= i6 i>('f[fe +9, ) ,+ Sl) < " 



(14.79) 



A change of variable to x = 
parameter £, 



r / 1 \ *i 

ctjp\ h 2 l and introduction of the 

£ = ^(l + 2 f (14.80) 

allows us to transform the integrals in A n (co) and A L {w) into the form: 
A n (<o) = "(- 2 + 02 ) \ " ^^P V&( z + *^ dx 
A x ((o) = ^ e(\ + 2 ) f °° exp [%g(z + i* 8 )] dx 



(14.81) 



* The fact that the limits of integration in (14.79) are t = ± oo may seem to contradict 
the approximations made in going from (14.76) to (14.77). The point is that for most 
frequencies the phase of the integrands in (14.79) oscillates very rapidly and makes the 
integrands effectively zero for times much smaller than those necessary to maintain the 
validity of (14.77). Hence the upper and lower limits on the integrals can be taken as 
infinite without error. Only for frequencies of the order of a> ~ (pip) ~ eo„ does the 
approximation fail. But we have seen in Section 14.4 that for relativistic particles essenti- 
ally all the frequency spectrum is at much higher frequencies. 



484 



Classical Electrodynamics 



The integrals in (14.81) are identifiable as Airy integrals, or alternatively 
as modified Bessel functions : 



I 



1 



x sin [f |(x + * x 3 )] dx = — K H (£) 
o V3 



i 



1 



cos [f i(x + |z 3 )] <*# = — K H (£) 
o 73 



(14.82) 



Consequently the energy radiated per unit frequency interval per unit 
solid angle is 



dljco) 
dQ 



3tt 2 c 



m 



+ 



Kk\S) + 



d 2 



(My 2 ) + & 



^m 



(14.83) 



The first term in the square bracket corresponds to radiation polarized in 
the plane of the orbit, and the second to radiation polarized perpendicular 
to that plane. 

We now proceed to examine this somewhat complex result. First we 
integrate over all frequencies and find that the distribution of energy in 
angle is 



f 

Jo 



dI((o) , 1 e 2 

dco = 



1 



dn 



16 P 



(i+*T 



1 + 



7(l/y 2 ) + 2 J 



(14.84) 



vy 



This shows the characteristic behavior seen in Section 14.3. Equation 
(14.84) can be obtained directly, of course, by integrating a slight generali- 
zation of the circular-motion power formula (14.44) over all times. As in 
(14.83), the first term in (14.84) corresponds to polarization parallel to the 
orbital plane, and the second to perpendicular polarization. Integrating 
over all angles, we find that seven times as much energy is radiated with 
parallel polarization as with perpendicular polarization. The radiation 
from a relativistically moving charge is very strongly, but not completely, 
polarized in the plane of motion. 

The properties of the modified Bessel functions summarized in (3.103) 
and (3.104) show that the intensity of radiation is negligible for £ > 1. 
From (14.80) we see that this will occur at large angles; the greater the 
frequency, the smaller the critical angle beyond which there will be 
negligible radiation. This shows that the radiation is largely confined to 
the plane containing the motion, as shown by (14.84), being more so 
confined the higher the frequency relative to dp. If to gets too large, 
however, we see that | will be large at all angles. Then there will be 
negligible total energy emitted at that frequency. The critical frequency 



[Sect. 14.6] 



Radiation by Moving Charges 



485 



co c beyond which there is negligible radiation at any angle can be denned 
by £ = 1 for = 0. Then we find 



-*(-H(-J- e 



(14.85) 



This critical frequency is seen to agree with our qualitative estimate (14.50) 
of Section 14.4. If the motion of the charge is truly circular, then cjp is 
the fundamental frequency of rotation, co . Then we can define a critical 
harmonic frequency to c = n c co , with harmonic number, 



\m<rf 



(14.86) 



Since the radiation is predominantly in the orbital plane for y > 1, it is 
instructive to evaluate the angular distribution (14.83) at = 0. For 
frequencies well below the critical frequency (co < co c ), we find 



dl{co) 



da 



e=o 



?]W 



For the opposite limit of w > co c , the result is 



dI((o) 



dQ 



e=o 



jL £_ v 2 — p -2o>/<»c 
2-7T C 



CO _. 

co„ 



(14.87) 



(14.88) 



These limiting forms show that the spectrum at 6 = increases with 
frequency roughly as co 2/3 well below the critical frequency, reaches a 
maximum in the neighborhood of co c , and then drops exponentially to 
zero above that frequency. 

The spread in angle at a fixed frequency can be estimated by determining 
the angle C at which £(0 C ) ~ 1(0) +1. In the low-frequency range 
(a> < co c ), £(0) is very small, so that £(0 C ) — 1- This g ives 



\cop' y\co' 



(14.89) 



We note that the low-frequency components are emitted at much wider 
angles than the average, <0 2 )^ ~ y" 1 . In the high-frequency limit 
(o> > co c ), 1(0) is large compared to unity. Then the intensity falls off in 
angle approximately as 



dI(co) dI(co) 



-3<oy 2 2 /(o e 



dQ. 



dQ. 



(14.90) 



9 = 



486 Classical Electrodynamics 

Thus the critical angle, defined by the \\e point, is 



y \3o)/ 



(14.91) 



This shows that the high-frequency components are confined to an angular 
range much smaller than average. Figure 14.10 shows qualitatively the 
angular distribution for frequencies small compared with, of the order of, 
and much larger than, co c . The natural unit of angle yd is used. 

The frequency distribution of the total energy emitted as the particle 
passes by can be found by integrating (14.83) over angles: 



I(a>) 



J-v/i 



dI(co) 



cos d6 ~2tt 



f 



dl(m) 



dd 



(14.92) 



t/2 dQ. J-°° dQ 

(remember that 6 is the colatitude). We can estimate the integral for the 
low-frequency range by using the value of the angular distribution (14.87) 
at 6 — and the critical angle 6 C (14.89). Then we obtain 

dI((o) 



I(ca>) ~ 2tt6 c 



dQ. 



0=0 C\ C 



(14.93) 



showing that the spectrum increases as co 1 - 4 for a> < a> c . This gives a very 

broad, flat spectrum at frequencies below (o c . For the high-frequency limit 

where oj > eo c we can integrate (14.90) over angles to obtain the reasonably 

accurate result, 2 , ^ 

I(co) "JlZr-yl—) e- 2(o/o " (14.94) 

c \a> c / 

A proper integration of (14.83) over angles yields the expression, 

_ 2 f °° 

I(co) = 2^3 - y — K, A {x) dx (14.95) 

c a> P J%(oi<o e 



dl(oi) 

da 




o>«co c 



yd ^ 



Fig. 14.10 Differential frequency 
spectrum as a function of angle. 
For frequencies comparable to 
the critical frequency a> c , the radi- 
ation is confined to angles of the 
order of y -1 . For much smaller 
(larger) frequencies, the angular 
spread is larger (smaller). 



[Sect. 14.6] 



Radiation by Moving Charges 



487 



2.0 



1 1 — I l I l I 



T 1 1 I II I I 




CO/Wc 



Fig. 14.11 Synchrotron radiation spectrum (energy radiated per unit frequency 

interval) as a function of frequency. The intensity is measured in units of ye 2 /c, while 

the frequency is expressed in units of co c (14.85). 



In the limit co < co c this reduces to the form (14.93) with a numerical 
coefficient 3.25, while for co > eo c it is equal to (14.94). The behavior of 
I(co) as a function of frequency is shown in Fig. 14.11. The peak intensity 
is of the order of e^yjc, and the total energy is of the order of e 2 ycoJc = 
3e 2 y 4 //>. This is in agreement with the value of 47re 2 y 4 /3/o for the radiative 
loss per revolution (14.32) in circular accelerators. 

The radiation represented by (14.83) and (14.95) is called synchrotron 
radiation because it was first observed in electron synchrotrons (1948). 
The theoretical results are much older, however, having been obtained for 
circular motion by Schott (1912). For periodic circular motion the 
spectrum is actually discrete, being composed of frequencies which are 
integral multiples of the fundamental frequency co Q = c/p. Since the 
charged particle repeats its motion at a rate of c\2ttp revolutions per second, 
it is convenient to talk about the angular distribution of power radiated 
into the nth multiple of co instead of the energy radiated per unit frequency 
interval per passage of the particle. To obtain the harmonic power 
expressions we merely multiply 1(a)) (14.95) or dI(oi)jdQ. (14.83) by the 
repetition rate cjlirp to convert energy to power, and by co = cjp to 
convert per unit frequency interval to per harmonic. Thus 



da 277-v 

P n = — y-JI((o = nco ) 



dljco) 
dQ 



(14.96) 



488 Classical Electrodynamics 

These results have been compared with experiment in some detail. * For 
this purpose it is necessary to average the spectra over the acceleration 
cycle of the machine, since the electron's energy increases continually (see 
Problem 14.13). With 80 Mev maximum energy, the spectrum extends 
from the fundamental frequency of co ^ 10 9 sec -1 to co c ~ 10 16 sec -1 , or 
A ^ 1700 angstroms. The radiation covers the visible region and is bluish 
white in color. Careful measurements are in full agreement with theory. 
Synchrotron radiation has been observed in the astronomical realm 
associated with sunspots, the Crab nebula, and perhaps the ^ dO 3 Mc/sec 
radiation from Jupiter. For the Crab nebula the radiation spectrum 
extends over a frequency range from radiofrequencies into the extreme 
ultraviolet, and shows very strong polarization. From detailed observa- 
tions it can be concluded that electrons with energies ranging up to 10 12 ev 
are emitting synchrotron radiation while moving in circular or helical 
orbits in a magnetic induction of the order of 10~ 4 gauss (see Problem 
14.15). The radio emission from Jupiter apparently comes from electrons 
trapped in Van Allen belts at distances several radii from Jupiter's surface. 
Whether these are relativistic electrons emitting synchrotron radiation, or 
nonrelativistic electrons emitting so-called cyclotron radiation as they 
spiral in the planet's magnetic field, is not clear at present. In any event, 
the radiation is strongly polarized parallel to the equator of Jupiter, as 
expected for particles trapped in a dipole field and spiraling around lines 
of force. 



14.7 Thomson Scattering of Radiation 

If a plane wave of monochromatic electromagnetic radiation is incident 
on a free particle of charge e and mass m, the particle will be accelerated 
and so emit radiation. This radiation will be emitted in directions other 
than that of the incident plane wave, but for nonrelativistic motion of the 
particle it will have the same frequency as the incident radiation. The 
whole process may be described as scattering of the incident radiation. 

The instantaneous power radiated by a particle of charge e in non- 
relativistic motion is given by Larmor's formula (14.21), 

4E. = J- & S in 2 (14.97) 

dQ 4ttc 3 

where © is the angle between the observation direction and the accelera- 
tion. The acceleration is provided by the incident plane electromagnetic 

* F. R. Elder, R. V. Langmuir, and H. C. Pollock, Phys. Rev., 74, 52 (1948); and 
especially D. H. Tomboulain and P. L. Hartman, Phys. Rev., 102, 1423 (1956). 



[Sect. 14.7] Radiation by Moving Charges 489 

wave. If the propagation vector is k, and the polarization vector c, the 
electric field can be written 

E(x, = cV^-*"" (14.98) 

Then, from the force equation for nonrelativistic motion, we have the 
acceleration, 

v(f) = c - E J k -*- iwt (14.99) 

m 

If we assume that the charge moves a negligible part of a wavelength 
during one cycle of oscillation, the time average of v 2 is |Re (v • v*). Then 
the average power per unit solid angle can be expressed as 

%)=t E 46Y sin * & (14100) 

Since the process is most simply viewed as a scattering, it is convenient to 
introduce a scattering cross section, defined by 

da Energy radiated/unit time/unit solid angle f14 10n 

dQ. ~~ Incident energy flux in energy/unit area/unit time 

The incident energy flux is just the time-averaged Poynting's vector for 
the plane wave, namely, c \E \ 2 IStt. Thus from (14.100) we obtain the 
differentia] scattering cross section, 

^= l-^-) 2 sin 2 (14.102) 

dQ \mcV 

If the wave is incident along the z axis with its polarization vector making 
an angle of tp with the x axis, as shown in Fig. 14.12, the angular distri- 
bution is 

sin 2 = 1- sin 2 cos 2 (<f> - y) (14.103) 

For unpolarized radiation the cross section is given by averaging over the 
angle ip. Thus 

Kl + cos 2 0) (14.104) 



da_ = |V \ 2 
dQ. \mcV 



This is called the Thomson formula for scattering of radiation by a free 
charge, and is appropriate for the scattering of X-rays by electrons or 
gamma rays by protons. The angular distribution is as shown in Fig. 14.13 
by the solid curve. The total scattering cross section, called the Thomson 
cross section, is 0/2 \? 

°t = t (—. : )' (14.105) 

3 \mc / 



490 



Classical Electrodynamics 



The Thomson cross section is equal to 0.665 x 10~ 24 cm 2 for electrons. The 
unit of length, e 2 /mc 2 = 2.82 x 10~ 13 cm, is called the classical electron 
radius, since a classical distribution of charge totaling the electronic 
charge must have a radius of this order if its electrostatic self-energy is to 
equal the electron mass (see Chapter 17). 

The classical Thomson result is valid only at low frequencies. For 
electrons quantum-mechanical effects enter importantly when the frequency 
a> becomes comparable to mc 2 /h, i.e, when the photon energy hen is 
comparable with, or larger than, the particle's rest energy mc 2 . Another 
way of looking at this criterion is that we expect quantum effects to appear 
if the wavelength of the radiation is of the order of, or smaller than, the 
Compton wavelength fr/mc of the particle. At these higher frequencies 
the angular distribution becomes peaked in the forward direction as shown 
in Fig. 14.13 by the dotted curves, always having, however, the Thomson 
value at zero degrees. The total cross section falls below the Thomson 
cross section (14.105). The process is then known as Compton scattering, 
and for electrons is described theoretically by the Klein-Nishina formula. 
For reference purposes we quote the asymptotic forms of the total cross 
section, as given by the Klein-Nishina formula : 

'8tt/ _ 2hco 



(e 2 Y 
\mcv 



+ 



mc 



hco <^ mc 2 



mc 
hco 



. (ihco 
- \mc 2 



+ 



(14.106) 



hco ;> mc 2 



For protons the departures from the Thomson formula occur at photon 




Fig. 14.12 



[Sect. 14.8] 



Radiation by Moving Charges 



491 




Fig. 14.13 Differential scattering cross section of unpolarized radiation by a free 
electron. The solid curve is the classical Thomson result. The dotted curves are given 
by the quantum-mechanical Klein-Nishina formula. The numbers on the curves refer 

to values of hco/mc 2 . 

energies above about 100 Mev. This is far below the critical energy 
hoi ~ Mc 2 ~ 1 Bev which would be expected in analogy with the electron 
Compton effect. The reason is that a proton is not a point particle like 
the electron with nothing but electromagnetic interactions, but is a complex 
entity having a spread-out charge distribution with a radius of the order of 
0.8 x 10~ 13 cm caused by strong interactions with pi mesons. The 
departure (a rapid increase in cross section) from Thomson scattering 
occurs at photon energies of the order of the rest energy of the pi meson 
(140 Mev). 



14.8 Scattering of Radiation by Quasi-free Charges; Coherent and 
Incoherent Scattering 



In the scattering of X-rays by atoms the angular distribution (14.104) is 
observed at wide angles, at least in light elements. But in the forward 
direction the scattering per electron increases rapidly to quite large values 
compared to the Thomson cross section. The reason is a coherent addition 
of the amplitudes from all electrons. From (14.18) it can be seen that the 



492 Classical Electrodynamics 

radiation field from a number of free charged particles will be 

E « = - Z e >\ D (14.107) 

C *—> L ff . Jret 

j J 

With (14.99) for the acceleration of the typical particle, we find 



, = ->x(nxc)p 
c- *—i m. 



exp 



E„ = 



/k • x,- — iw I f ^ 

c / J 



R< 



(14.108) 



In calculating the radiation it is sufficient to approximate 7?, in the 
exponent by the form (14.63). Then, in complete analogy with the steps 
from (14.97) to (14.102), we find the scattering cross section, 



da 
d~Q 



where 



2— 2 A " 



q = — n — k 

c 



sin 2 



(14.109) 



(14.110) 



is the vectorial change in wave number in the scattering. 

Equation (14.109) applies to free charged particles instantaneously at 
positions x,. Electrons in atoms, for example, are not free. But if the 
frequency of the incident radiation is large compared to the characteristic 
frequencies of binding, the particles can be treated as free while being 
accelerated by a pulse of finite duration. Thus (14.109) can be applied to 
the scattering of high-frequency (compared to binding frequencies) radia- 
tion by bound charged particles. The only thing that remains before 
comparison with experiment is to average (14.109) over the positions of all 
the particles in the bound system. Thus the observable cross section for 
scattering is 



da 
dQ. 



J*-*, 



sin 







(14.111) 



where the symbol < > means average over all possible values of x 3 . 

The cross section (14.111) shows very different behavior, depending on 
the value of |q|. The coordinates x,- have magnitudes of the order of the 
linear dimensions of the bound system. If we call this dimension a, then 
the behavior of the cross section is very different in the two regions, 
qa < 1 and qa > 1 . If the scattering angle is 6, the magnitude of q is 
2k sin (6/2). Thus the dividing line between the two domains occurs for 



[Sect. 14.8] Radiation by Moving Charges 493 

angles such that 

2kasin-~l (14.112) 

2 

If the frequency is low enough so that ka < 1, then the limit qa < 1 will 
apply at all angles. But for frequencies where ka > 1, there will be a region 
of forward angles less than 

c ~i- (14.113) 

ka 

where the limit qa < 1 holds, and a region of wider angles where the limit 
qa > 1 applies. 

For qa < 1, the arguments of exponents in (14.1 1 1) are all so small that 
the exponential factors can be approximated by unity. Then the differential 
cross section becomes 



.. da 
hm — = 

aa->0 dQ. 



S -^- 2 sin 2 = Z 2 (—J sin 2 (14.114) 

^-i m.c 2 \mc 2 / 



where the last form is appropriate for electrons in an atom of atomic 
number Z. This shows the coherent effect of all the particles, giving an 
intensity corresponding to the square of the number of particles times the 
intensity for a single particle. 

In the opposite limit of qa > 1 the arguments of the exponents are large 
and widely different in value. Consequently the cross terms in the square 
of the sum will average to zero. Only the absolute squared terms will 
survive. Then the cross section takes the form : 

Hm S = y (—J sm2 = z (A \f sm2 ° ( i4 - ii5 > 

i 
where again the final form is for electrons in an atom. This result corre- 
sponds to the incoherent superposition of scattering from the individual 
particles. 

For the scattering of X-rays by atoms the critical angle (14.113) can be 
estimated, using (13.95) as the atomic radius. Then one finds the numerical 
value, 

6 C ~ — (14.116) 

hco (kev) 

For angles less than 6 C the cross section rises rapidly to a value of the order 
of (14.114), while at wide angles it is given by Z times the Thomson result, 
(14.115), or for high-frequency X-rays or gamma rays by the Klein- 
Nishina formula, shown in Fig. 14.13. 



494 



Classical Electrodynamics 



14.9 Cherenkov Radiation 

A charged particle in uniform motion in a straight line in free space does 
not radiate. But a particle moving with constant velocity through a material 
medium can radiate if its velocity is greater than the phase velocity of light 
in the medium. Such radiation is called Cherenkpv radiation, after its 
discoverer, P. A. Cherenkov (1937). The emission of Cherenkov radiation 
is a cooperative phenomenon involving a large number of atoms of the 
medium whose electrons are accelerated by the fields of the passing particle 
and so emit radiation. Because of the collective aspects of the process it 
is convenient to use the macroscopic concept of a dielectric constant e 
rather than the detailed properties of individual atoms. 

A qualitative explanation of the effect can be obtained by considering 
the fields of the fast particle in the dielectric medium as a function of time. 
We denote the velocity of light in the medium by c and the particle velocity 
by v . Figure 14.14 shows a succession of spherical field wavelets for v < c 
and for v > c. Only for v > c do the wavelets interfere constructively to 
form a wake behind the particle. The normal to the wake makes an angle 




u< c 




Fig. 14.14 Cherenkov radiation. Spherical wavelets of fields of a particle traveling 
less than, and greater than, the velocity of light in the medium. For v > c, an electro- 
magnetic "shock" wave appears, moving in the direction given by the Cherenkov 

angle d c . 



[Sect. 14.9] Radiation by Moving Charges 495 

6 C with the velocity direction, where 



cos 6 C = - (14.117) 

v 



This is the direction of emission of the Cherenkov radiation. 

Although we have already found the fields appropriate to the Cherenkov- 
radiation problem in Section 13.4, and have even given an expression 
(13.82) for the energy emitted as Cherenkov radiation, it is instructive to 
look at the problem from the point of view of the Lienard-Wiechert 
potentials. We will make use of Section 13.4 to the extent of noting that, 
for a nonpermeable medium, we may discuss the fields and energy radiated 
as if the particle moved in free space with a velocity v > c, provided at the 
end of the calculation we make the replacements, 

c < , «,-- L (14.118) 

where e is the dielectric constant.* We will simplify the analysis by 
assuming that e is independent of frequency. But our final results will be 
for individual frequency components and so will be easily generalized. 

For a point charge in arbitrary motion the Lienard-Wiechert potentials 
were obtained in Section 14. 1 . It was tacitly assumed there that the particle 
velocity was less than the velocity of light. Then the potentials (14.6) at a 
given point in space-time depended on the behavior of the particle at one 
earlier point in space-time, the retarded position. This situation corre- 
sponds in the left side of Fig. 14.14 to the fact that a given point lies on 
only one circle. When v > c, however, we see from the right side of the 
figure that two retarded positions contribute to the field at a given point in 
space-time. The scalar potential in (14.6) is replaced by 



$(x, = e 



1 



+ e 



-kRji LkRJ2 



1 



(14.119) 



where the indices 1 and 2 indicate the two retarded times t{ and t 2 '. 

To determine the two times t{ and t 2 ' we consider the vanishing of the 
argument of the delta function in (14.3): 

t > + ' x ~ r ^ -t = (14.120) 

c 

* From (13.54) it is evident that we are dealing in this way with potentials <X>' = Ve®, 
A' = A, and fields E' = VcE, B' = B. Then, for example, Poynting's vector is 

S' = — (E' X B') -^ — C —- (E' XB') = S 

4tt 4ttV€ 



496 



Classical Electrodynamics 




Fig. 14.15 



For a particle with constant velocity v, we can take r(t') = yf. With 
X = (x — vt) as the vector distance from the present position P' of the 
particle to the observation point P, (14.120) becomes 



(t - t') = - |X + y(t - t')\ 



The solution of this quadratic is 



(t - f) = 



-X • v ± V(X . v) 2 - (v 2 - c 2 )X 2 
(u 2 - c 2 ) 



(14.121) 



(14.122) 



Only roots that are real and positive have physical meaning. For v < c, 
the square root is real and larger than |X • v|. Hence there is only one 
valid root for (t — t'), as already noted. But for v > c, there are other 
possibilities. First we note that even when the square root is real (as it is for 
directions more or less parallel or antiparallel to v) it is smaller in magni- 
tude than |X • v|. Consequently there is no root for (t — t') when X and v 
have an acute angle between them; the fields do not get ahead of the 
particle. If a is the angle between X and v, as shown in Fig. 14.15, we see 
furthermore that the square root is imaginary for cos 2 a < [1 — (c 2 /v 2 )]. 
But for backward angles, such that 



cos 






< a < 77 



there are two real, positive values of (t — t') as solutions of (14.121). Thus 
the potentials exist only inside the Cherenkov cone defined by cos a = 
-[1 -(c 2 !v 2 )p. 



[Sect. 14.9] Radiation by Moving Charges 491 

The values of kR corresponding to the two roots (14.122) are easily 

shown to be 

K R = T - [(X • v) 2 - (v* - c 2 )X 2 ] H (14.123) 

c 

Actually in (14.119) the absolute values are required, because of the sign 
inherent in the Jacobian derivative in (14.4). Thus the two terms add, and 
the potentials can be written 

<D(x, = 



Xy/l - (v 2 lc 2 ) sin 2 a 
A(x, = -^>(x, 



(14.124) 



c 

These potentials are valid inside the Cherenkov cone, become singular on 
its surface, and vanish outside the cone. They represent a wave front 
traveling at velocity c in the direction d c (14.1 17). The singularity is not a 
physical reality, of course. It comes from our assumption that the velocity 
of light in the medium is independent of frequency. For high enough 
frequencies (short enough wavelengths) the phase velocity of light in the 
medium will approach the velocity of light in vacuo. This variation with 
frequency will cause a smoothing at short distances which will eliminate 
singularities. 

The potentials (14.124) are special cases of the potentials whose Fourier 
transforms are given by (13.57). The fields which can be found from 
(14.124) are similarly the Fourier transforms of the fields (13.64) and 
(13.65), assuming e(eo) is a real constant. The calculation of energy 
radiated proceeds exactly as in Section 13.4 with the integration of the 
Poynting's vector over a cylinder, as in (13.71), yielding the final expression 
(13.82) for energy radiated per unit distance. 

The discussion presented so far, with the appearance of a Cherenkov 
"shock wave" for v > c, is the proper macroscopic description of the 
origin of Cherenkov radiation. If, however, one is interested only in the 
angular and frequency distribution of the radiation and not in the mecha- 
nism, it is possible to give a simple, nonrigorous derivation, using the 
substitutions ( 1 4. 1 1 8). The angular and frequency distribution of radiation 
emitted by a charged particle in motion is given by (14.67): 

dI(co) e 2 co 2 If 00 , , to ('- 5 T Q ) 

— ^-^ = n x (n x \)e \ c ' 

For a particle moving in a nonpermeable, dielectric medium transfor- 
mation (14.118) yields 

— ^—^ = e^ n x (n x \)e \ ° ' dt (14.126) 

dQ. 47r 2 c 3 I J-°° 



(14.125) 



498 Classical Electrodynamics 

For a uniform motion in a straight line, r(t) = vt. Then we obtain 



dl(w) 



n x v 



o>_ f°° i<oth 

1lT J -oo 



dQ. c 

The integral is a Dirac delta function. Then 
dI(co) e 2 e A p 2 sin 2 



v ) dt 



dQ. 



1 6(1 - e /2 /? cos 0)| s 



(14.127) 



(14.128) 



where 6 is measured relative to the velocity v. The presence of the delta 
function guarantees that the radiation is emitted only at the Cherenkov 
angle 6 C : 

cos 6 C = -y (14.129) 

The presence of the square of a delta function in angles in (14.128) means 
that the total energy radiated per unit frequency interval is infinite. This 
infinity occurs because the particle has been moving through the medium 
forever. To obtain a meaningful result we assume that the particle passes 
through a slab of dielectric in a time interval IT. Then the infinite integral 
in (14.127) is replaced by 

jo f y e «(i - g") dt = ^ Sin ^ r(1 - ^ C ° S °>3 (14.130) 
2ttJ-t tt [coT(1 - e A p cos 0)] 

The absolute square of this function peaks sharply at the angle 6 C , provided 
coT > 1. Assuming that /? > —^ , so that the angle d c exists, the integral 
over angles is 

'^/ojiy sin>[a>T(l-^cosOX = 2joT 
\ rr 1 [coT(l - e*P COS 0)] 2 fie* 

showing that the amount of radiation is proportional to the time interval. 
From (14.128) we find that the total energy radiated per unit frequency 
interval in passing through the slab is 



/■ 



1(c) = tE S i n 2 Q c ( 2cj 8T) 
c 2 



(14.132) 



This can be converted into energy radiated per unit frequency interval per 
unit path length by dividing by 2cfiT. Then, with (14. 129) for 6 C , we obtain 



dI((o) 
dx 



e co 

^2 



1 



1 



P 2 e(co)J 



(14.133) 



where co is such that e(co) > (l//? 2 ), in agreement with (13.82). 



[Sect. 14.9] 



Radiation by Moving Charges 



499 



€(W) 




Fig. 14.16 Cherenkov band. 

Radiation is emitted only in 

shaded frequency range where 

e(co) > jS- 2 . 



The properties of Cherenkov radiation can be utilized to measure 
velocities of fast particles. If the particles of a given velocity pass through 
a medium of known dielectric constant e, the light is emitted at the 
Cherenkov angle (14.129). Thus a measurement of the angle allows deter- 
mination of the velocity. Since the dielectric constant of a medium in 
general varies with frequency, light of different colors is emitted at some- 
what different angles. Figure 14.16 shows a typical curve of e(eo), with a 
region of anomalous dispersion at the upper end of the frequency interval. 
The shaded region indicates the frequency range of the Cherenkov light. 
Since the dielectric medium is strongly absorbent at the region of anom- 
alous dispersion, the escaping light will be centered somewhat below the 
resonance. Narrow band filters may be employed to select a small interval 
of frequency and so improve the precision of velocity measurement. For 
very fast particles (£ < 1) a gas may be used to provide a dielectric 
constant differing only slightly from unity and having (e — 1) variable 
over wide limits by varying the gas pressure. Counting devices using 
Cherenkov radiation are employed extensively in high-energy physics, as 
instruments for velocity measurements, as mass analyzers when combined 
with momentum analysis, and as discriminators against unwanted slow 
particles. 



REFERENCES AND SUGGESTED READING 

The radiation by accelerated charges is at least touched on in all electrodynamics 
textbooks, although the emphasis varies considerably. The relativistic aspects are treated 
in more or less detail in 

Iwanenko and Sokolow, Sections 39-43, 

Landau and Lifshitz, Classical Theory of Fields, Chapters 8 and 9, 

Panofsky and Phillips, Chapters 18 and 19, 

Sommerfeld, Electrodynamics, Sections 29 and 30. 
Extensive calculations of the radiation emitted by relativistic particles, anticipating many 
results rederived in the period 1940-1950, are presented in the interesting monograph by 

Schott. 



500 Classical Electrodynamics 

The scattering of radiation by charged particles is presented clearly by 
Landau and Lifshitz, Classical Theory of Fields, Sections 9.11-9.13, and 
Electrodynamics of Continuous Media, Chapters XIV and XV. 



PROBLEMS 

14.1 Verify by explicit calculation that the Lienard-Wiechert expressions for all 
components of E and B for a particle moving with constant velocity agree 
with the ones obtained in the text by means of a Lorentz transformation. 
Follow the general method at the end of Section 14.1. 

14.2 Using the Lienard-Wiechert fields, discuss the time-average power radiated 
per unit solid angle in nonrelativistic motion of a particle with charge e, 
moving 

(a) along the z axis with instantaneous position z(t) = a cos co t, 

(b) in a circle of radius R in the x-y plane with constant angular 
frequency co . 

Sketch the angular distribution of the radiation and determine the total 
power radiated in each case. 

14.3 A nonrelativistic particle of charge ze, mass m, and kinetic energy E makes 
a head-on collision with a fixed central force field of finite range. The 
interaction is repulsive and described by a potential V(r), which becomes 
greater than E at close distances. 

(a) Show that the total energy radiated is given by 

N 2 Jr mill 



3 ra 2 c 3 . 



dV 
dr 



Vv(r m m) - V{r) 



where r m in is the closest distance of approach in the collision. 

(b) If the interaction is a Coulomb potential V(r) = zZe 2 jr, show that 
the total energy radiated is 



AlV = 



zmv 6 



45 Zc 3 



where v is the velocity of the charge at infinity. 
14.4 A particle of mass m, charge q, moves in a plane perpendicular to a 
uniform, static, magnetic induction B. 

(a) Calculate the total energy radiated per unit time, expressing it in 
terms of the constants already defined and the ratio y of the particle's 
total energy to its rest energy. 

(b) If at time t = the particle has a total energy E = y Q m&, show 
that it will have energy E = ymc 2 < E at a time t, where 

f _ 3m 3 c 5 /I _ 1 
_ 2(fB 2 \y Yo) 
provided y > 1 . 

(c) If the particle is initially nonrelativistic and has a kinetic energy € 
at t = 0, what is its kinetic energy at time tl 

id) If the particle is actually trapped in the magnetic dipole field of the 
earth and is spiraling back and forth along a line of force, does it radiate 



[Probs. 14] 



Radiation by Moving Charges 



501 



14.5 



14.6 



14.7 



14.8 



more energy while near the equator, or while near its turning points? 
Why? Make quantitative statements if you can. 

As in Problem 14.2a a charge e moves in simple harmonic motion along 
the z axis, z(t') = a cos (oo t'). 
{a) Show that the instantaneous power radiated per unit solid angle is: 

dP(t') _ e 2 cft 4 sin 2 6 cos 2 (cot') 
dQ. 477a 2 (1 + p cos sin co t'f 

where P = acojc. 

(b) By performing a time averaging, show that the average power per 
unit solid angle is : 

JD „2„/?4 r A i R2 „™2 A 1 

sin 2 

(c) Make rough sketches of the angular distribution for nonrelativistic 
and relativistic motion. 

Show explicitly by use of the Poisson sum formula or other means that, 
if the motion of a radiating particle repeats itself with periodicity T, the 
continuous frequency spectrum becomes a discrete spectrum containing 
frequencies that are integral multiples of the fundamental. Show that a 
general expression for the power radiated per unit solid angle in each 
multiple m of the fundamental frequency co = 2tt \T is: 



dP 


e 2 cp 


4 + p 2 cos 2 e 


dCl 


327ra 2 


_(i - p 2 cos 2 eyA_ 



dP m 
dCl 



? 2 (o^m 2 



1 ' 



v(/) x nexp 



4-^)]"' 



(a) Show that for the simple harmonic motion of a charge discussed in 
Problem 14.5 the average power radiated per unit solid angle in the mth 
harmonic is : 

dP e 2 cB 2 

ii§ = S m% tan2 QJmKm ^ cos 6) 

(b) Show that in the nonrelativistic limit the total power radiated is all 
in the fundamental and has the value: 



2e 2 
3-3^ 



4„2 



where a 2 is the mean square amplitude of oscillation. 

A particle of charge e moves in a circular path of radius R in the x-y plane 

with constant angular velocity o> . 

(a) Show that the exact expression for the angular distribution of power 
radiated into the mth multiple of oo is : 



dPrn 
dCl 



e 2 <o 4 i? 2 

27TC 3 



,n dJ m (mP sin d) \ 2 
\ d(mP sin 6) J 



cot 2 6 
+ -35- Jm\mP sin0) 



where p = co Q Rlc, and J m (x) is the Bessel function of order m. 

(b) Assume nonrelativistic motion and obtain an approximate result for 
dP m /dCl. Show that the results of Problem 14.26 are obtained in this limit. 

(c) Assume extreme relativistic motion and obtain the results found in 
the text for a relativistic particle in instantaneously circular motion. 
(Watson, pp. 79, 249, may be of assistance to you.) 



502 Classical Electrodynamics 

14.9 Bohr's correspondence principle states that in the limit of large quantum 
numbers the classical power radiated in the fundamental is equal to the 
product of the quantum energy (Ha> ) and the reciprocal mean lifetime of 
the transition from principal quantum number n to (n — 1). 

(a) Using nonrelativistic approximations, show that in a hydrogen-like 
atom the transition probability (reciprocal mean lifetime) for a transition 
from a circular orbit of principal quantum number n to (n — 1) is given 
classically by 

1 = 2£7Z£YwcM 
t 3 fic\ he J h n b 

(b) For hydrogen compare the classical value from (a) with the correct 
quantum-mechanical results for the transitions 2p -+ Is (1.6 x 10 _9 sec), 
4/^3^/(7.3 x 10- 8 sec), 6/* —5^ (6.1 x 10- 7 sec). 

14.10 Periodic motion of charges gives rise to a discrete frequency spectrum in 
multiples of the basic frequency of the motion. Appreciable radiation in 
multiples of the fundamental can occur because of relativistic effects 
(Problems 14.7 and 14.8) even though the components of velocity are truly 
sinusoidal, or it can occur if the components of velocity are not sinusoidal, 
even though periodic. An example of this latter motion is an electron 
undergoing nonrelativistic elliptic motion in a hydrogen atom. 

The orbit can be specified by the parametric equations 

x — a(cos u — e) 



y = aV\ — e 2 sin u 
where 

co t = u — c sin u 

a is the semimajor axis, e is the eccentricity, co is the orbital frequency, 
and u is the angle in the plane of the orbit. In terms of the binding energy 
B and angular momentum L, the various constants are 



e 2 L 2BL* 8B S 

IB' V me 4 ' ° me* 

(a) Show that the power radiated in the &th multiple of co is 



*-£ (to Hp 



(J k '(ke)f + \l—f\j k \ke) j 



where J k (x) is a Bessel function of order k. 

(b) Verify that for circular orbits the general result (a) agrees with part 
(a) of Problem 14.9. 

14.11 Instead of a single charge e moving with constant velocity co in a circular 
path of radius R, as in Problem 14.8, a set of N such charge moves with 
fixed relative positions around the same circle. 

(a) Show that the power radiated into the wth multiple of co is 

dPJJN) dPJX) F fm 
d£l dCl m{ } 



[Probs. 14] Radiation by Moving Charges 503 

where dP m {\)ldO. is the result of part (a) in Problem 14.8, and 



F m (N) 



N 



2^ 

i=i 



Bj being the angular position of they'th charge at t = t . 

(b) Show that, if the charges are uniformly spaced around the circle, 
energy is radiated only into multiples of Na> , but with an intensity N 2 
times that for a single charge. Give a qualitative explanation of these facts. 

(c) Without detailed calculations show that for nonrelativistic motion 
the dependence on N of the total power radiated is dominantly as fi 2N , so 
that in the limit N -»■ oo no radiation is emitted. 

(d) By arguments like those of (c) show that for relativistic particles the 
radiated power varies with N mainly as exp (— 27V/3y 3 ) for N > y 3 , so 
that again in the limit N -+ <x> no radiation is emitted. 

(e) What relevance have the results of (c) and (d) to the radiation 
properties of a steady current in a loop ? 

14.12 As an idealization of steady-state currents flowing in a circuit, consider a 
system of N identical charges q moving with constant speed v (but subject 
to accelerations) in an arbitrary closed path. Successive charges are 
separated by a constant small interval A. 

Starting with the Lienard-Wiechert fields for each particle, and making 
no assumptions concerning the speed v relative to the velocity of light, 
show that, in the limit N -* oo, q ->0, and A ->-0, but Nq = constant 
and q/A = constant, no radiation is emitted by the system and the electric 
and magnetic fields of the system are the usual static values. 

(Note that for a real circuit the stationary positive ions in the conductors 
will produce an electric field which just cancels that due to the moving 
charges.) 

14.13 Assume that the instantaneous power spectrum radiated by an electron in 
a synchrotron is given by 

P(o>, t) ~ - - y{t) (—) %-*»!<», 

77 P \€0j 

where co c = 3to y 3 (/). 

(a) If the electrons increase their energy approximately linearly during 
one cycle of operation, show that the power spectrum, averaged over one 
cycle of operation, is 

(P(co, /)> ^-t Vm& yA r^dy 

3* p Jx y A 

where x = 2o>/a> cmax 

(b) Determine limiting forms for the spectrum when x < 1 and x > 1 . 

(c) By finding tables of the integral (it is an incomplete gamma function) 
or by graphical integration for x = 0. 1 , 0.5, 1 .0, 1.5, determine numerically 
the spectrum, plot it as a function of log [ft>/eo cmax ], and compare it with 
the curves given by Elder, Langmuir, and Pollock, Phys. Rev., 74, 52 
(1948), Fig. 1. 

14.14 (a) Within the framework of approximations of Section 14.6, show that, 
for a relativistic particle moving in a path with instantaneous radius of 
curvature />, the frequency-angle spectra of radiations with positive and 



504 Classical Electrodynamics 

negative helicity are 
dl 



w-um+'l 



*w«)±7; —Knit) 



(b) From the formulas of Section 14.6 and (a) above, discuss the 
polarization of the total radiation emitted as a function of frequency and 
angle. In particular, determine the state of polarization at (1) high 
frequencies (co > co c ) for all angles, (2) intermediate and low frequencies 
(co < co c ) for large angles, (3) intermediate and low frequencies at very 
small angles. 

(c) See the paper by P. Joos, Phys. Rev. Letters, 4, 558 (1960), for 
experimental comparison. 

14.15 Consider the synchrotron radiation from the Crab nebula. Electrons with 
energies up to at least 10 12 ev move in a magnetic field of the order of 
10~ 4 gauss. 

(a) For E = 10 12 ev, B = 10 -4 gauss, calculate the orbit radius p, the 
fundamental frequency a> = c/p, and the critical frequency co c . 

(b) Show that for a relativistic electron of energy £ in a constant 
magnetic field the power spectrum of synchrotron radiation can be written 



tf'te) 



P(E, co) = const 

where f(x) is a cutoff function having the value unity at * = and 
vanishing rapidly for x > 1 [e.g.,/ ~ exp ( -2co/co c ), as in Problem 14.13], 
and to c = (3eB/mc)(Elmc 2 ) 2 cos 6, where 6 is the pitch angle of the helical 
path. 

(c) If electrons are distributed in energy according to the spectrum 
N(E) dE ~ E~ n dE, show that the synchrotron radiation has the power 
spectrum 

<P(co)> dco ~ co~ a dco 
where a = (« — l)/2. 

(d) Observations on the radiofrequency and optical continuous 
spectrum from the Crab nebula show that on the frequency interval from 
co ~ 10 8 sec -1 to co ~ 6 x 10 15 sec -1 the constant a ~ 0.35. At higher 
frequencies the spectrum of radiation falls steeply with a ^ 1.5. Determine 
the index n for the electron-energy spectrum, as well as an upper cutoff for 
that spectrum. Is this cutoff consistent with the numbers of part (a) ? 

0) From the result of Problem 14.46 find a numerical value for the 
time taken by an electron to decrease in energy from infinite energy to 
10 12 ev in a field of 10 -4 gauss. How does this compare with the known 
lifetime of the Crab nebula ? 
14.16 Assuming that Plexiglas or Lucite has an index of refraction of 1.50 in the 
visible region, compute the angle of emission of visible Cherenkov 
radiation for electrons and protons as a function of their energies in Mev. 
Determine how many quanta with wavelengths between 4000 and 6000 
angstroms are emitted per centimeter of path in Lucite by a 1-Mev 
electron, a 500-Mev proton, a 5-Bev proton. 



15 



Bremsstrahlung , 

Method of Virtual Quanta, 

Radiative Beta Processes 



In Chapter 14 radiation by accelerated charges was discussed in a 
general way, formulas were derived for frequency and angular distributions, 
and examples of radiation by both nonrelativistic and relativistic charged 
particles in external fields were treated. The present chapter is devoted to 
problems of emission of electromagnetic radiation by charged particles in 
atomic and nuclear processes. 

Particles passing through matter are scattered and lose energy by 
collisions, as described in detail in Chapter 13. In these collisions the 
particles undergo acceleration ; hence they emit electromagnetic radiation. 
The radiation emitted during atomic collisions is customarily called 
bremsstrahlung (braking radiation) because it was first observed when high- 
energy electrons were stopped in a thick metallic target. For nonrelativistic 
particles the loss of energy by radiation is negligible compared with the 
collisional energy loss, but for ultrarelativistic particles radiation can be 
the dominant mode of energy loss. Our discussion of bremsstrahlung and 
related topics will begin with the nonrelativistic, classical situation. Then 
semiclassical arguments will be used, as in Chapter 13, to obtain plausible 
quantum-mechanical modifications. Relativistic effects, which produce 
significant changes in the results, will then be presented. 

The creation or annihilation of charged particles is another process in 
which radiation is emitted. Such processes are purely quantum mechanical 
in origin. There can be no attempt at a classical explanation of the 
basic phenomena. But given that the process does occur, we may legiti- 
mately ask about the spectrum and intensity of electromagnetic radiation 

505 



506 Classical Electrodynamics 

accompanying it. The sudden creation of a fast electron in nuclear beta 
decay, for example, can be viewed for our purposes as the violent accelera- 
tion of a charged particle initially at rest to some final velocity in a very 
short time interval, or, alternatively, as the sudden switching on of the 
charge of the moving particle in the same short interval. We will discuss 
nuclear beta decay and orbital-electron capture in these terms in Sections 
15.7 and 15.8. 

In radiation problems, such as the emission of bremsstrahlung or radia- 
tive beta decay, the wave nature of the charged particles involved produces 
quantum-mechanical modifications very similar to those appearing in our 
earlier energy-loss considerations. These can be taken into account in a 
relatively simple way. But there is a more serious deficiency which occurs 
only in radiation problems. It is very difficult to take into account the 
effects on the trajectory of the particle of the energy and momentum carried 
off by radiation. This is not only because radiation reaction effects are 
relatively hard to include (see Chapter 17), but also because of the discrete 
quantum nature of the photons emitted. Thus, even when modifications 
are made to describe the quantum-mechanical nature of the particles, our 
results are limited in validity by the restriction that the emitted photon have 
an energy small compared to the total energy available. At the upper end 
of the frequency spectrum our semiclassical expressions will generally have 
only qualitative validity. 



15.1 Radiation Emitted during Collisions 

If a charged particle makes a collision, it undergoes acceleration and 
emits radiation. If its collision partner is also a charged particle, they both 
emit radiation and a coherent superposition of the radiation fields must be 
made. Since the amplitude of the radiation fields depends (nonrelativisti- 
cally) on the charge times the acceleration, the lighter particle will radiate 
more, provided the charges are not too dissimilar. In many applications 
the mass of one collision partner is much greater than the mass of the other. 
Then for the emission of radiation it is sufficient to treat the collision as the 
interaction of the lighter of the two particles with a fixed field of force. We 
will consider only this situation, and will leave more involved cases to the 
problems at the end of the chapter. 

From (14.65) we see that a nonrelatlvistic particle with charge e and 
acceleration c(3(f) radiates energy with an intensity per unit solid angle per 
unit frequency interval, 

dI(co) e 2 [ , . Ut-™™) 2 



[Sect. 15.1] Bremsstrahlung, Virtual Quanta, Radiative Beta Processes 507 

The position vector r(/) has the order of magnitude (v)t, relative to a suit- 
able origin, where (v) is a typical velocity of the problem. This means that 
the second term in the exponential in (15.1) is of the order of (v)/c times 
the first. For nonrelativistic motion, it can be neglected. Its neglect is 
sometimes called the dipole approximation, in analogy with the multipole 
expansion of Section 9.2. Then we find the approximate expression, 



dljco) = e 2 
dO. 4tt 2 c 



I 



n x (n x $)e imt dt 



(15.2) 



If we consider a collision process, the acceleration caused by the field of 
force exists only for a limited time t, namely, the collision time : 



r~- (15.3) 

v 



where a is a characteristic distance over which the force is appreciable. 
Then the integral in (15.2) is over a time interval of order t. This means 
that t provides a natural parameter with which to divide the frequencies of 
the radiation emitted into low frequencies (cot < 1) and high frequencies 
(cot > 1). In the low-frequency limit, the exponential in (15.2) is sensibly 
constant over the period of acceleration. Then the integration can be 
performed immediately : 

(p(t)e i(at dt ~ |p(0 dt = p 2 - Pi = Ap (15.4) 

where c$ x and c(J 2 are the initial and final velocities, and A(S is the 
vectorial change. Then the energy radiated is 

^- } ~ — | API 2 sin 2 0, cot < 1 (15.5) 

dn 4tt 2 c ^ 

where is measured relative to the direction of A (5. The total energy 
radiated per unit frequency interval in this limit is 

I(co) ~ — -| API 2 , cot < 1 (15.6) 

3tt c 

In the high-frequency limit (cot > 1) the exponential in (15.2) oscillates 
very rapidly compared to the variation of $(t) in time. Consequently the 
integrand has a very small average value, and the energy radiated is 
negligible. The frequency spectrum will appear qualitatively as shown in 
Fig. 15.1. It will be convenient sometimes to make the approximation that 



508 Classical Electrodynamics 

the spectrum is given by a step function : 

' 2e 2 
/(*>) = 



-I API- a»<l 



0, COT > 1 

For a single encounter with a definite A|3 this is not a very good approxi- 
mation, but if an average over many collisions with various A (3 is wanted 
the approximation is adequate. 

The angular distribution (15.5) includes all polarizations of the emitted 
radiation. Sometimes it is of interest to exhibit the intensity for a definite 
state of polarization. In collision problems it is usual that the direction of 
the incident particle is known and the direction of the radiation is known, 
but the deflected particle's direction, and consequently that of A (3, are not 
known. Consequently the plane containing the incident beam direction 
and the direction of the radiation is a natural one with respect to which 
one specifies the state of polarization of the radiation. 

For simplicity we consider a small angle deflection so that A (3 is approxi- 
mately perpendicular to the incident direction. Figure 15.2 shows the 
vectorial relationships. Without loss of generality n, the observation 
direction, is chosen in the x-z plane, making an angle 6 with the incident 
beam. The change in velocity A (3 lies in the x-y plane, making an angle cf> 
with the x axis. Since the direction of the scattered particle is not observed, 
we will average over j>. The unit vectors €,, and e ± are polarization vectors 
parallel and perpendicular to the plane containing (3 X and n. 

The direction of polarization of the radiation is given by the vector 
n x (n x A (3). This is perpendicular to n (as it must be) and can be 
resolved into components along e,, and e ± . Thus 

n x (n x A(3) = A0[cos d cos <j> e„ — sin <f> € ± ] (15.8) 

The absolute squares of the components in (15.8), averaged over <f>, give 



-I(O)^\A0\ 




1/7 



Fig. 15.1 Frequency spectrum of radiation emitted in 
a collision of duration t with velocity change A/J. 



[Sect. 15.2] Bremsstrahlung, Virtual Quanta, Radiative Beta Processes 509 




Fig. 15.2 



the intensities of radiation for the two polarization states. The results are 



dD. 

dI L ((o) 
dQ. 



8tt 2 c 

_£!_ 
8tt 2 c 



= -VI Ap| 2 cos 2 



|A(J|< 



(15.9) 



These angular distributions are valid for all types of nonrelativistic small- 
angle collisions. They have been verified in detail for the continuous 
X-ray spectrum produced by electrons of kinetic energies in the kilovolt 
range. It is evident that the sum of the two intensities is consistent with 
(15.5) and yields a total radiated intensity equal to (15.6). 



15.2 Bremsstrahlung in Nonrelativistic Coulomb Collisions 

The most common situation where a continuum of radiation is emitted 
is the collision of a fast particle with an atom. As a model for this process 
we will consider first the collision of a fast, but nonrelativistic, particle of 
charge ze, mass M, and velocity v with a fixed point charge Ze. For 
simplicity we will assume that the deflection of the incident particle is small. 
Then the same arguments as were used in Chapter 13 on the limits of impact 
parameters will be involved. In fact, much of the discussion can be trans- 
planted bodily from the treatment of energy loss. 

For a small deflection in a point Coulomb field of charge Ze the 
momentum change is transverse and is given by (13.1) times Z. Thus the 
net change of velocity of the incident particle passing at impact parameter 
b has the magnitude 



Ay = 



2zZe 2 
Mvb 



(15.10) 



510 Classical Electrodynamics 

The frequency spectrum will be given approximately by (15.7) (times z 2 ) 
with a collision time (15.3) r~ b/v. Thus the spectrum extends from 

co = to comax ^ v/b : 

' 8 (z 2 e 2 \ 2 Z 2 e 2 
37rWc 2 / c 



/(co, b) ~ 



0, 



1 

ft 2 ' 


V 
CO < — 

ft 




co > - 
ft 



(15.11) 



Just as in the energy-loss process, the useful physical quantity is a cross 
section obtained by integrating over all possible impact parameters. 
Accordingly we define the radiation cross section %{co), with dimensions 
(area-energy/frequency), 

x(co) = J /(co, ft)277-ft db (15.12) 

The classical limits on the impact parameters can be found by arguments 
analogous to those of Section 13.1. The classical minimum impact param- 
eter is [see(13.5)-(13.7)]: 



^ (c) _ zZe 2 
Mv 2 



ftmin = . (15.13) 



while the maximum value is governed by the cutoff in the spectrum (15. 1 1). 
If we fix our attention on a given frequency co in %(co), it is evident that only 
for impact parameters less than 

ftmax^- (15.14) 

ft) 

will the accelerations be violent and rapid enough to produce significant 
radiation at that frequency. With these limits on ft the radiation cross 
section is 

where A is a numerical factor of the order of unity which takes into account 
our uncertainties in the exact limits on impact parameters. This result is 
valid only for frequencies where the argument of the logarithm is large 
compared to unity, corresponding to ft max > ft min . This means that there 
is a classical upper limit co ( ^ x to the frequency spectrum given by 

co£L (15.16) 

zZe* 

For highly charged, massive, slow particles the classical radiation cross 
section will be valid, but just as in the energy-loss phenomena the wave 



[Sect. 15.2] Bremsstrahlung, Virtual Quanta, Radiative Beta Processes 511 

nature of particles enters importantly for lightly charged, swift particles. 
The quantum modifications are very similar to those discussed in Section 
13.3. The wave nature of the incident particle sets a quantum-mechanical 
lower limit on the impact parameters, 

b8L~— (15.17) 

Mv 

This means that the radiation cross section is approximately 

instead of (15.15). We note that the arguments of the logarithm differ by 
the factor r\ (13.42) (times Z to give the product of the charges). The same 
rules about domains of validity of the classical and quantum-mechanical 
formulas apply here as for the energy loss. The frequency spectrum of the 
quantum cross section extends up to a maximum frequency co^ x of the 
order of 

a>SL~^ (15.19) 

n 

We note that this is approximately the conservation of energy limit, 
Wmax = Mv 2 /2h. Since the classical result holds only when r\ > 1, we see 
that 

a&L ^ " a&L < *>mL (15-20) 

This shows that the classical frequency spectrum is always confined to very 
low frequencies compared to the maximum allowed by conservation of 
energy. Thus the classical domain is of little interest. In what follows we 
will concentrate on the quantum-mechanical results. 

Although the upper limit (15.19) is in rough accord with conservation 
of energy, the quantum radiation cross section has only qualitative validity 
at the upper end of the frequency spectrum. As was discussed in the 
introduction to this chapter, the reason is the discrete quantum nature of 
the photons emitted. For soft photons with energies far from the maximum 
the discrete nature is unimportant because the energy and momentum 
carried off are negligible. But for hard photons near the end point of the 
spectrum the effects are considerable. One obvious and plausible way to 
include the conservation of energy requirement is to argue that the impact 
parameters (15.14) and (15.17) should involve an average velocity, 

(v) = K»* + »/) = "7= (y/E + V^M (15-21) 



512 



Classical Electrodynamics 





1 


1 1 1 


l'|i 


1 










- 


A 








- 


3 






Sv -» v> ^ Semiclassical 


- 


X 




V Classical 


s. ^Z" quantum 
Bethels. 


---> 






V^tj = 10 


Heitler ^v 


— 




1 


1 Nsj 


1 


- 



0.2 



0.4 



0.6 



0.8 



1.0 



■ flu 
E 



Fig. 15.3 Radiation cross section (energy x area/unit frequency) for Coulomb 
collisions as a function of frequency in units of the maximum frequency (E/h). The 
classical spectrum is confined to very low frequencies. The curve marked "Bethe- 
Heitler" is the quantum-mechanical Born approximation result, while the "semi- 
classical quantum" curve is (15.18). 

where E = \Mv 2 is the initial kinetic energy of the particle, and ho is the 
energy of the photon emitted. With this average velocity in place of v in 
(15.18), we obtain 



X<M — 



16 ZV 



f z 2 e 2 \ 2 (c\ 2 
\McV \v) 



I (JE + JE - hoSf 
-2 ha) 



(15.22) 



If X = 2, this cross section is exactly the quantum-mechanical result in 
the Born approximation, first calculated by Bethe and Heitler (1934). The 
argument of the logarithm evidently equals unity when hoy = \Mv l , so 
that the conservation of energy requirement is properly satisfied. Figure 
15.3 shows the shape of the radiation cross section as a function of 
frequency. The Bethe-Heitler formula (15.22) is compared with our 
classical and semiclassical quantum formulas (15.15) and (15.18) with A = 2 
and r\ = 10. 

The bremsstrahlung spectrum is sometimes expressed as a cross section 
for photon emission with dimensions of area/unit photon energy. Thus 



hojo hiems (hco) d(hco) = %{(jo) dco 



(15.23) 



[Sect. 15.3] Bremsstrahlung, Virtual Quanta, Radiative Beta Processes 513 

The bremsstrahlung photon cross section is evidently 

, t , 16ZV/zV\ 2 /c\ 2 ln( ) n ._ 

. brems (M^ T — (—)(-) ^^ (15.24) 

where the argument of the logarithm is that of (15.15) or (15.22). Since the 
logarithm varies relatively slowly with photon energy, the main dependence 
of the cross section on photon energy is as (/ko) _1 , known as the typical 
bremsstrahlung spectrum. 

The radiation cross section %((o) depends on the properties of the particles 
involved in the collision as Z¥/A/ 2 , showing that the emission of radiation 
is most important for electrons in materials of high atomic number. The 
total energy lost in radiation by a particle traversing unit thickness of 
matter containing N fixed charges Ze (atomic nuclei) per unit volume is 

Af . /''"max 

™* = N %{a>)da> (15.25) 

dx Jo 

Using (15.22) for %{(o) and converting to the variable of integration 
x = (ha)/E), we can write the radiative energy loss as 

*&« = 16 NZ (Zl)^ p in (l±JT=*\ dx (15>26) 

dx 3 \Hc/Mc 2 Jo \ y/z / 

The dimensionless integral has the value unity. For comparison we write 
the ratio of radiative energy loss to collision energy loss (13.13) or (13.44): 

rf£rad_. 4^mjVj 2 J_ (152?) 



i — i 



dEcoii 3tt 137 M\c! In J3, 

For nonrelativistic particles (v < c) the radiative loss is completely negli- 
gible compared to the collision loss. The fine structure constant (e 2 /hc = 
1/137) enters characteristically whenever there is emission of radiation as 
an additional step beyond the basic process (here the deflection of the 
particle in the nuclear Coulomb field). The factor m\M appears because 
the radiative loss involves the acceleration of the incident particle, while 
the collision loss involves the acceleration of an electron. 



15.3 Relativistic Bremsstrahlung 

For relativistic particles making collisions with atomic nuclei there are 
characteristic modifications in the radiation emitted. Our first thought 
would be that the nonrelativistic discussion of the previous sections would 
not be valid at all, and that a full relativistic treatment would be necessary. 



514 



Classical Electrodynamics 








Laboratory 
system 



Coordinate 
system K' 



Fig. 15.4 Radiation emitted during relativistic collisions viewed from the laboratory 
(nucleus at rest) and the frame K' (incident particle essentially at rest). 

But it is one of the great virtues of the special theory of relativity (aside 
from being correct and necessary) that it allows us to choose a convenient 
reference frame for our calculation and then transform to the laboratory at 
the end. Thus we will find that all but the final steps of the relativistic 
bremsstrahlung calculation can be done nonrelativistically. 

There are two aspects. First of all, we know that radiation emitted by a 
highly relativistic particle is confined to a narrow cone of half-angle of the 
order of Mc 2 /E, where E is its total energy. Thus, unless we are interested 
in very fine details, it is sufficient to consider only the total energy radiated 
at a given frequency. The second point is that except for very close 
collisions the incident particle is deflected only slightly in an encounter and 
loses only a very small amount of energy. In the reference frame K', where 
the incident particle is at rest initially and the nucleus moves by with 
velocity v ~ c, the corresponding motion of the incident particle is non- 
relativistic throughout the collision. This means that in the frame K' the 
radiation process can be treated entirely nonrelativistically. The connection 
between the radiation process as viewed in the laboratory and in the 
coordinate frame K' is sketched in Fig. 15.4. 

Almost all the arguments previously presented in Sections 15.1 and 15.2 
apply. The only modifications are in the limits on the impact parameters. 
The relativistic contraction of the fields (see Section 11.10) makes the col- 
lision time (11.120) smaller by a factor y = E/Mc 2 . This means that the 
maximum impact parameter is increased from (15.14) to 



h ~Z? 

f max — — 



(15.28) 



where a>' is the emitted frequency in the coordinate system K'. The 
minimum impact parameter for these radiation problems is not the 
expected hjp = fr/yMv, even though this is the magnitude of "smearing 



[Sect. 15.3] Bremsstrahlung, Virtual Quanta, Radiative Beta Processes 515 

out" of the particle due to quantum effects. The proper value is still (15.17), 
without factors of y, as can be seen from the following argument. In the 
emission of radiation all parts of an extended charge distribution must 
experience the same acceleration at the same time. Otherwise interference 
effects will greatly reduce the amount of radiation. This means that 
appreciable radiation will occur only if the width of the pulse of accelera- 
tion due to the passing nuclear field is large compared to the "smearing 
out" of the charge. The pulse width is of the order of b/y, while the 
transverse smearing-out distance is of the order of hjyMv. This sets a 
lower limit on impact parameters equal to (15.17), even for relativistic 
motion. With (15.11), (15.12), and these revised impact parameters, the 
radiation cross section %'{co') in the system K' is 



16 ZV fW \ 2 (cY ln UyMv 2 \ 
\McV V n \ hco' I 



«^~(tt;I (-) ln r-^J ( 15 - 29 > 



To transform this result to the (unprimed) laboratory frame we need to 
know the transformation properties of the radiation cross section and the 
frequency. The radiation cross section has the dimensions of (cross- 
sectional area) • (energy) • (frequency) -1 . Since energy and frequency 
transform in the same way under Lorentz transformations, while transverse 
dimensions are invariant, the radiation cross section is a Lorentz invariant: 

X (a>) = X '(co') (15.30) 

The transformation of frequencies is according to the relativistic Doppler- 
shift formula (11.38): 

co = yco'(l + P cos 0') (15.31) 

where 6' is the angle of emission in the frame K'. The cross section %'(co') 
is the total cross section, integrated over angles in K '. Since the accelera- 
tion is predominantly transverse in that frame, the radiation is emitted 
essentially symmetrically about 6' = tt/2. Consequently on the average 
we have co = yco '.* With this substitution for co' in (15.29) we obtain the 
radiation cross section in the laboratory: 

The only change from the nonrelativistic result (15.18) is the factor y 2 in 
the argument of the logarithm. Conservation of energy requires that this 

* This result can be obtained from the original transformation (11.37), a>' = 
yco(l — j8 cos 8), by noting that, for y ;> 1 and 6 <: 1, we have co' ~ (co/2y)(l + y 2 6 2 ). 
Since the average value of y 2 6 2 in the laboratory is of the order of unity, we obtain 

co' ~ co/y. 



516 Classical Electrodynamics 

expression be used only for frequencies such that < two < (y — \)Mc 2 
^ yMc 2 . We note that quanta with laboratory energies in the range 
Mc 2 < hoi < yMc 2 come from quanta with ha>' < Mc 2 in the transformed 
frame K'. 

The above derivation of x(°>) in the laboratory is somewhat casual in that the 
dependence of transformed frequency on angle was not treated rigorously. We 
should actually consider the differential cross section in the frame K': 



dCl' 



UyMv 2 \ " 

I hot' J _ 



— (1 + cos 2 0') 

I6rr 



(15.33) 



where A is the coefficient of the logarithm in (15.32). The square-bracketed 
angular factor comes from the sum of the two terms in (15.9) and is normalized 
to a unit integral over solid angles. When transformed according to (11.38), 
(15.33) becomes in the laboratory 

*-> - Al, j 2 ": W ,Jl *' + £*? (15.34) 



dQ. \H(o(l +y 2 d 2 )j2n (1 + y 2 2 ) 4 

The angular distribution is peaked sharply in the forward direction. The angular 
factor falls off as (yd)-* for yd > 1. Of course, (15.34) is not valid for angles 
0^1. But the order of magnitude is correct, the intensity of radiation being a 
factor y~ 4 smaller at backward angles than in the forward direction, and 
approaching the limiting value (at = n) : 

,. d X ((o) 3 A^ UMv 2 \ , 1cr , c . 

Since almost all the radiation is confined to angles 6 <; 1 , we may approximate 
the solid angle element dQ, ~ 2tt0 dd = (w/y 2 ) d(y 2 6 2 ), and integrate over the 
interval < y 2 2 < oo with little error. This yields the total radiation cross 
section, 

which differs insignificantly from the previous result (15.32). 



(15.36) 



15.4 Screening Effects; Relativistic Radiative Energy Loss 

In the treatment of bremsstrahlung so far we have ignored the effects of 
the atomic electrons. As direct contributors to the acceleration of the 
incident particle they can be safely ignored, since their contribution per 
atom is of the order of Z _1 times the nuclear one. But they have an indirect 
effect through their screening of the nuclear charge. The potential energy 
of the incident particle in the field of the atom can be approximated by the 
form (13.94). This means that there will be negligible radiation emitted 
for collisions at impact parameters greater than the atomic radius (13.95). 



[Sect. 15.4] Bremsstrahlung, Virtual Quanta, Radiative Beta Processes 517 

We can include this approximately in our previous calculations by denning 
a maximum impact parameter due to screening by the atomic electrons, 

b&*~a~lAJ& (15.37) 

Then we must use the smaller of the two values (15.28) and (15.37) for 
6max in the argument of the logarithm. The ratio is 

frmax _ 192 /v\ hoi (15 38) 

frmax~ZMc/yW 

where m is the electronic mass and we have used the average value 
oi = (x>\y. This shows that for low enough frequencies the screening value 
is always smaller than (15.28). The limiting frequency co s below which we 

must use 6^ ax is, in units of the spectrum end point w m ax = -(y — l)Mc 2 , 



Wmax 



■ ?L(»\ y lz±lY ^ 

192 \M/\y - 1/ 



2Z H m /c\ 
192 M\vJ 
Z m 

192 M 7 



(15.39) 



where the upper (lower) line is the nonrelativistic (relativistic) limiting 
form. When a> < co s the argument of the logarithm in the radiation cross 
section (15.32) becomes independent of frequency: 

*as~jl»2*!H (15.40) 

6min Z m C 

This makes the radiation cross section approach the constant value, 

for co < w s . Then the energy radiated per unit frequency interval at low 
frequencies is finite, rather than logarithmically divergent. This is the same 
type of effect as the screening produces in making the small-angle scattering 
(13.96) finite, rather than divergent as 0~ 4 for a pure Coulomb field. 

Except for extremely low velocities the screening frequency co s is very 
small compared to eo ma x in the nonrelativistic limit. A typical figure is 
s /fi) m3X ~ 0.07 for electrons of 100-Kev kinetic energy incident on a gold 
target (Z = 79). For heavier nonrelativistic particles the ratio is even 
smaller. This means that the spectrum shown in Fig. 15.3 is altered only 
at very low frequencies for nonrelativistic bremsstrahlung. 



518 Classical Electrodynamics 




Fig. 15.5 Radiation cross section 
in the complete screening limit. 
The constant value is the semi- 
classical result. The curve marked 
"Bethe-Heitler" is the quantum- 
mechanical Born approximation. 

For extremely relativistic particles the screening can be "complete." 
Complete screening occurs when co s > eomax. This occurs at energies 
greater than the critical value, 

_ /192M 



E s=\^r 1 ) Mc ( 15 - 42 > 



For electrons, E s ~ 42 Mev in aluminum (Z = 13) and 23 Mev in lead 
(Z = 82). The corresponding values for mu mesons are 2 x 10 6 Mev and 
10 6 Mev. Because of the factor Mjm, screening is important only for 
electrons. When E > E s , the radiation cross section is given by the 
constant value (15.41) for all frequencies. Figure 15.5 shows the radiation 
cross section (15.41) in the limit of complete screening, as well as the 
corresponding Bethe-Heitler result. Their proper quantum treatment 
involves a slowly varying factor which changes from unity at co = to 
0.75 at co = wmax. For cosmic-ray electrons and for electrons from most 
high-energy electron accelerators, the bremsstrahlung is in the complete 
screening limit. Thus the photon spectrum shows a typical (/zco) -1 
behavior. 

The radiative energy loss was considered in the nonrelativistic limit in 
Section 15.2 and was found to be negligible compared to the energy loss 
by collisions. For ultrarelativistic particles, especially electrons, this is no 
longer true. The radiative energy loss is given approximately in the limit 
y > 1 by 

d£rad 



dx 



^i N m^\[ M "\j^\ dm (15 .43) 

3 c \Mc 2 / Jo \b m J 



where the argument of the logarithm depends on whether co < a> s or 
ft» > co s . For negligible screening (co s < com&x) we find approximately 



WW 



^_^ Ar ^ L( ^_ | ln(Ay)yMc2 (1544) 

dx 3 nc 



[Sect. 15.4] Bremsstrahlung, Virtual Quanta, Radiative Beta Processes 519 
For higher energies where complete screening occurs this is modified to 



dx 



L6 N 5V(f!fl) 2 ln (^M)l yM ^ (15.45) 

.3 He \McV \Z A m) V 



showing that eventually the radiative loss is proportional to the particle's 
energy. 
The comparison of radiative loss to collision loss now becomes 

( l\92M \ 

iE™>~± 111) n n \z 1A ™> (15 46) 

d£coii 37r\137/M \nB a 

The value of y for which this ratio is unity depends on the particle and on 
Z. For electrons it is y ~ 200 for air and y ^ > 20 for lead. At higher 
energies, the radiative energy loss is larger than the collision loss and for 
ultrarelativistic particles is the dominant loss mechanism. 

At energies where the radiative energy loss is dominant the complete 
screening result (15.45) holds. Then it is useful to introduce a unit of 
length X Q , called the radiation length, which is the distance a particle travels 
while its energy falls to e _1 of its initial value. By conservation of energy, 
we may rewrite (15.45) as 

d_E__E_ 
dx X 

with solution, E(x) = E e~ x,x ° (15.47) 

where the radiation length X is 



X n = 



_ 3 he \Mc 2 ) n I 



X192M 

Z Vz m 



(15.48) 



For electrons, some representative values of X are 32 gm/cm 2 (270 meters) 
in air at NTP, 19 gm/cm 2 (7.2 cm) in aluminum, and 4.4 gm/cm 2 (0.39 cm) 
in lead.* In studying the passage of cosmic-ray or man-made high-energy 
particles through matter, the radiation length X is a convenient unit to 
employ, since not only the radiative energy loss is governed by it, but also 
the production of negaton-positon pairs by the radiated photons, and so 
the whole development of the electronic cascade shower. 

* These numerical values differ by <~20-30 per cent from those given by Rossi, p. 55, 
because he uses a more accurate coefficient of 4 instead of 16/3 and Z(Z +1) instead of 
Z 2 in (15.48). 



520 



Classical Electrodynamics 



15.5 Weizsacker-Williauis Method of Virtual Quanta 

The emission of bremsstrahlung and other processes involving the 
electromagnetic interaction of relativistic particles can be viewed in a way 
that is very helpful in providing physical insight into the processes. This 
point of view is called the method of virtual quanta. It exploits the similarity 
between the fields of a rapidly moving charged particle and the fields of a 
pulse of radiation (see Section 11.10) and correlates the effects of the 
collision of the relativistic charged particle with some system with the 
corresponding effects produced by the interaction of radiation (the virtual 
quanta) with the same system. The method was developed independently 
by C. F. Weizsacker and E. J. Williams in 1934. 

In any given collision there are an "incident particle" and a "struck 
system." The perturbing fields of the incident particle are replaced by an 
equivalent pulse of radiation which is analyzed into a frequency spectrum 
of virtual quanta. Then the effects of the quanta (either scattering or 
absorption) on the struck system are calculated. In this way the charged- 
particle interaction is correlated with the photon interaction. The table 
lists a few typical correspondences and specifies the incident particle and 

Correspondences between charged particle interactions and photon interactions 

Incident Struck 

Particle Process Particle System Radiative Process bmm 



Bremsstrahlung in 
electron (light 
particle)-nucleus 
collision 


Nucleus 


Electron 

(light 

particle) 


Scattering of virtual hjMv 
photons of nuclear 
Coulomb field by the 
electron (light particle) 


Collisional ioniza- 
tion of atoms (in 
distant collisions) 


Incident 
particle 


Atom 


Photoejection of atomic a 

electrons by virtual 

quanta 


Electron disintegra- 
tion of nuclei 


Electron 


Nucleus 


Photodisintegration 
of nuclei by virtual 
quanta 


Larger 
of 


Production of pions 
in electron-nuclear 
collisions 


Electron 


Nucleus 


Photoproduction of 
pions by virtual 
quanta interactions 
with nucleus 


*■ hlymv 
and 
R 



struck system. From the table we see that the struck system is not always 
the target in the laboratory. For bremsstrahlung the struck system is the 



[Sect. 15.5] Bremsstrahlung, Virtual Quanta, Radiative Beta Processes 521 




Pi 



>■ 



I I 



P 2 



*3 



Fig. 15.6 Relativistic charged particle passing the struck system S and the equivalent 

pulses of radiation. 

lighter of the two collision partners, since its radiation scattering power is 
greater. For bremsstrahlung in electron-electron collision it is necessary 
from symmetry to take the sum of two contributions where each electron 
in turn is the struck system at rest initially in some reference frame. 

The chief assumption in the method of virtual quanta is that the effects 
of the various frequency components of equivalent radiation add inco- 
herently. This will be true provided the perturbing effect of the fields is 
small, and is related to our assumption in Section 15.2 that the struck 
particle moves only slightly during the collision. 

The spectrum of equivalent radiation for an incident particle of charge 
q, velocity v ~ c, passing a struck system S at impact parameter b, can be 
found from the fields of Section 11.10: 



yb 



£ i (f) -V+yW) 
B 2 (t) = 0^(0 
£ 3 (0 = -q 



yvt 



(15.49) 



(b 2 + y W)" J 



For j8 ~ 1 the fields JE^f) and B 2 {t) are completely equivalent to a pulse of 
plane polarized radiation P x incident on S in the x 3 direction, as shown in 
Fig. 15.6. There is no magnetic field to accompany E 3 (t) and so form a 
proper pulse of radiation P 2 incident along the x x direction, as shown. 
Nevertheless, if the motion of the charged particles in S is nonrelativistic 
in this coordinate frame, we can add the necessary magnetic field to create 
the pulse P 2 without affecting the physical problem because the particles 
in S respond only to electric forces. Even if the particles in S are influenced 
by magnetic forces, the additional magnetic field implied by replacing E 3 (t) 
by the radiation pulse P 2 is not important, since the pulse P 2 will be seen 
to be of minor importance anyway. 



522 



Classical Electrodynamics 



From the discussion of Section 14.5, especially equations (14.51), (14.52), 
and (14.60), it is evident that the equivalent pulse P x has a frequency 
spectrum (energy per unit area per unit frequency interval) I x {oj, b) given 
by 



Jifo b) = ±- lE^atf 



(15.50) 



where E x (co) is the Fourier transform (14.54) of E\(i) in (15.49). Similarly 
the pulse P z has the frequency spectrum, 



I 2 (co, b) = ± \E 3 (oj)\* 
2tt 



(15.51) 



The Fourier integrals have already been calculated in Chapter 13 and are 
given by (13.29) and (13.30). The two frequency spectra are 



Ii(o), b) 
I 2 (co, b) 



W 



77 C \V 



oM 2 K 2 (o>b 

yvi \yv. 



1 (2*\* ,/«* 



y \yv 






yv 



(15.52) 



We note that the intensity of the pulse P 2 involves a factor y~ 2 and so is of 
little importance for ultrarelativistic particles. The shapes of these spectra 
are shown qualitatively in Fig. 15.7. The behavior is easily understood if 
one recalls that the fields of pulse P x are bell-shaped in time with a width 
At ~ b/yv. Thus the frequency spectrum will contain all frequencies up 
to a maximum of order co ma x ~ 1/A*. On the other hand, the fields of 
pulse P 2 are similar to one cycle of a sine wave of frequency co ~ yv/b. 



1 <7 2 / \' 



2 Jl 

b 2 




Fig. 15.7 Frequency spectra of the two equivalent pulses of radiation. 



[Sect. 15.5] Bremsstrahlung, Virtual Quanta, Radiative Beta Processes 523 

Consequently its spectrum will contain only a narrow range of frequencies 
centered around yvjb. 

In collision problems we must sum the frequency spectra (15.52) over 
the various possible impact parameters. This gives the energy per unit 
frequency interval present in the equivalent radiation field. As always in 
such problems we must specify a minimum impact parameter 6 min . The 
method of virtual quanta will be useful only if b min can be so chosen that 
for impact parameters greater than Z> min the effects of the incident particle's 
fields can be represented accurately by the effects of equivalent pulses of 
radiation, while for small impact parameters the effects of the particle's 
fields can be neglected or taken into account by other means. Setting aside 
for the moment how we choose the proper value of 6 m i n , we can write 
down the frequency spectrum integrated over possible impact parameters, 

J(o>) = 2tt\ [I^co, b) + I 2 (a), b)]b db (15.53) 

•'bmin 

where we have combined the contributions of pulses P x and P 2 . This 
integral has already been done in Section 13.3, equation (13.35). The 
result is 



7T C W 



» 2 



xKJx)K x {x) - f-j x\K x \x) ~ K \x)) 



2d 1 



(15.54) 



where 



x = ^^ (15.55) 

yv 



For low frequencies (co < yvjbmi^) the energy per unit frequency interval 
reduces to 

i^^VL^-ll] (15.56, 

TV C W L \ COOmin ' 2c J 

whereas for high frequencies (co > yv/bmin) the spectrum falls off 
exponentially as 

/(co) - ~( £ ) 2 (l - —)e- i2<obmiM (15.57) 

Figure 15.8 shows an accurate plot of /(w) (15.54) for v ~ c, as well as the 
low-frequency approximation (15.56). We see that the energy spectrum 
consists predominantly of low-frequency quanta with a tail extending up to 
frequencies of the order of lyv/bmin. 



524 



/(o>) 

q 2 /irc 



Classical Electrodynamics 

— i 1 1 — i — i — tt~t 



1 - 



0.1 




Low-frequency 
approximation 



j i i '■' ' ' 



0.5 



Fig. 15.8 Frequency spectrum of virtual quanta for a relativistic particle, with the 
energy per unit frequency I(a>) in units of q 2 lrrc and the frequency in units of yv/b min . 
The number of virtual quanta per unit energy interval is obtained by dividing by h 2 co. 

The number spectrum of virtual quanta N(hco) is obtained by using the 
relation, 

I(co) dco = hctiN(hco) d(H(o) (15.58) 

Thus the number of virtual quanta per unit energy interval in the low- 
frequency limit is 

„2 



tt\Hc' W HcoL \ cobmin J 2c 



(15.59) 



The choice of minimum impact parameter & min must be considered. In 
bremsstrahlung, b m \n = hjMv, where M is the mass of the lighter particle, 
as discussed in Section 15.3. For collisional ionization of atoms, 6min c^ a, 
the atomic radius, closer impacts being treated as collisions between the 
incident particle and free electrons. In electron disintegration of nuclei or 
electron production of mesons from nuclei, b m ^ = hjyMv or 6 m in = R, 
the nuclear radius, whichever is larger. The values are summarized in the 
table on p. 520. 



[Sect. 15.6] Bremsstrahlung, Virtual Quanta, Radiative Beta Processes 525 

15.6 Bremsstrahlung as the Scattering of Virtual Quanta 

The emission of bremsstrahlung in a collision between an incident 
relativistic particle of charge ze and mass M and an atomic nucleus of 
charge Ze can be viewed as the scattering of the virtual quanta in the 
nuclear Coulomb field by the incident particle in the coordinate system K', 
where the incident particle is at rest. The spectrum of virtual quanta I(co') 
is given by (15.54) with q = Ze. The minimum impact parameter is HjMv, 
so that the frequency spectrum extends up to co' ~ yMc 2 fh. 

The virtual quanta are scattered by the incident particle (the struck 
system in K') according to the Thomson cross section (14.105) at low 
frequencies and the Klein-Nishina formula (14.106) at photon energies 
hco' > Mc 2 . Thus, in the frame K', for frequencies small compared to 
Mc 2 /h, the radiation cross section x'(co') is given by 

Z'M^f(g)V) (15-60) 

Since the spectrum of virtual quanta extends up to yMc 2 /h, the approxi- 
mation (15.56) can be used for I(co') in the region (co' < Mc 2 \h). Thus the 
radiation cross section becomes 



Ky ' 3 c \Mc 2 ' L \ hco' J 



(15.61) 



where extreme relativistic motion (v ^ c) has been assumed. 

This is essentially the same cross section as (15.29), and can be trans- 
formed to the laboratory in the same way as was done in Section 15.3. 
Equations (15.60) and (15.61), involving the Thomson cross section, are 
valid only for quanta in K' with frequencies co' < Mc 2 jh. For frequencies 
co' > Mc 2 \h, we must replace the constant Thomson cross section (14.105) 
with the quantum-mechanical Klein-Nishina formula (14.106), which falls 
off rapidly with increasing frequency. This shows that in K' the bremsstrah- 
lung quanta are confined to a frequency range < co' < Mc 2 /h, even 
though the spectrum of virtual quanta in the nuclear Coulomb field 
extends to much higher frequencies. The restricted spectrum in K' is 
required physically by conservation of energy, since in the laboratory 
system where co = yco' the frequency spectrum is limited to < co < 
(yMc 2 /h). A detailed treatment using the angular distribution of scattering 
from the Klein-Nishina formula yields a bremsstrahlung cross section in 
complete agreement with the Bethe-Heitler formulas (Weizsacker, 1934). 

The effects of screening on the bremsstrahlung spectrum can be dis- 
cussed in terms of the Weizsacker- Williams method. For a screened 



526 Classical Electrodynamics 

Coulomb potential the spectrum of virtual quanta is modified from (15.56). 
The argument of the logarithm is changed to a constant, as was discussed 
in Section 15.4. 

Further applications of the method of virtual quanta to such problems 
as collisional ionization of atoms and electron disintegration of nuclei are 
deferred to the problems at the end of the chapter. 

15.7 Radiation Emitted during Beta Decay 

In the process of beta decay an unstable nucleus with atomic number Z 
transforms spontaneously into another nucleus of atomic number (Z ± 1) 
while emitting an electron (=fe) and a neutrino. The process is written 
symbolically as 

Z->{Z± l) + e* +v (15.62) 

The energy released in the decay is shared almost entirely by the electron 
and the neutrino, with the recoiling nucleus getting a completely negligible 
share because of its very large mass. Even without knowledge of why or 
how beta decay takes place, we can anticipate that the sudden creation of a 
rapidly moving charged particle will be accompanied by the emission of 
radiation. As mentioned in the introduction, either we can think of the 
electron initially at rest and being accelerated violently during a short time 
interval to its final velocity, or we can imagine that its charge is suddenly 
turned on in the same short time interval. The heavy nucleus receives a 
negligible acceleration and so does not contribute to the radiation. 

For the purposes of calculation we can assume that at t = an electron 
is created at the origin with a constant velocity v = c$. Then, according 
to (14.67), the intensity distribution in frequency and angle of the radiation 
emitted is 

dI(co) e 2 co 2 f°° , ^ i0 >(t-±™\ , 2 
v - In x (n x P)e*T « ) dt 

Jo 



dO. 



\tt*C 



(15.63) 



Since (5 is constant, the position of the electron is r(t) = c$t. Then the 
intensity distribution is 



^ = £^> sin 2 r 

dQ. 477V Jo 



yico(l — p cos 6)t 



dt 



(15.64) 



where 6 is measured from the direction of motion of the emerging electron. 
Thus the angular distribution is 

dljoi) = _£!_ 02 sin 2 
dQ 4tt 2 c P (1 - cos Of U > 



[Sect. 15.7] Bremsstrahlung, Virtual Quanta, Radiative Beta Processes 527 



v(t) 




Fig. 15.9 



while the total intensity per unit frequency interval is 



/(co) = - 

7TC 



A In 



1-8/ J 



(15.66) 



For 8 < 1, (15.66) reduces to /(co) ^± 2e 2 8 2 /37rc, showing that for low- 
energy beta particles the radiated intensity is negligible. 

The intensity distribution (15.66) is a typical bremsstrahlung spectrum 
with number of photons per unit energy range given by 



N(ha)) 



= _e 2 _/j_ 



irhc \haiJ LB 



±ln 



1 + 
1 - 



-2 



(15.67) 



It sometimes bears the name "inner bremsstrahlung" to distinguish it from 
bremsstrahlung emitted by the same beta particle in passing through 
matter. It appears that the spectrum extends to infinity, thereby violating 
conservation of energy. We can obtain qualitative agreement with con- 
servation of energy by appealing to the uncertainty principle. Figure 15.9 
shows a qualitative sketch of the electron velocity as a function of time. 
Our calculation is based on a step function with the acceleration time t 
vanishingly small. From the uncertainty principle, however, we know that 
for a given uncertainty in energy AE the uncertainty in time At cannot be 
smaller than At ~ fi/AE. In the act of creation of the beta particle, 
AE = E = ymc 2 , so that the acceleration time t must be of the order of 
t ^ fr/E. When this is transformed into frequency, the well-known 
arguments show that the frequency spectrum will not extend appreciably 
beyond eo max ~ E\h, thereby satisfying the conservation of energy require- 
ment at least qualitatively. 

The total energy radiated is approximately 



£rad 



Jo 



I{pS) da> ^ 



e 
irhc 



.8 \\-BJ J 



(15.68) 



528 Classical Electrodynamics 

For very fast beta particles, the ratio of energy going into radiation to the 
particle energy is 



-Erad ^ 2 e 



2 



\mcv 



(15.69) 



This shows that the radiated energy is a very small fraction of the total 
energy released in beta decay, even for the most energetic beta processes 
Cdnax ^ 30mc 2 ). Nevertheless, the radiation can be observed, and pro- 
vides useful information for nuclear physicists. 

In the actual beta process the energy release is shared by the electron 
and the neutrino so that the electron has a whole spectrum of energies up 
to some maximum. Then the radiation spectrum (15.66) must be averaged 
over the energy distribution of the beta particles. Furthermore, a quantum- 
mechanical treatment leads to modifications near the upper end of the 
photon spectrum. These are important details for quantitative comparison 
with experiment. But the origins of the radiation and its semiquantitative 
description are given adequately by our classical calculation. 



15.8 Radiation Emitted in Orbital-Electron Capture— Disappearance of 
Charge and Magnetic Moment 

In beta emission the sudden creation of a fast electron gives rise to 
radiation. In orbital-electron capture the sudden disappearance of an 
electron does likewise. Orbital-electron capture is the process whereby an 
orbital electron around an unstable nucleus of atomic number Z is 
captured by the nucleus, transforming it into another nucleus with atomic 
number (Z — 1), with the simultaneous emission of a neutrino which 
carries off the excess energy. The process can be written symbolically as 

Z + e--*(Z-l) + v (15.70) 

Since a virtually undetectable neutrino carries away the decay energy if 
there is no radiation, the spectrum of photons accompanying orbital 
electron capture is of great importance in yielding information about the 
energy release. 

As a simplified model we consider an electron moving in a circular 
atomic orbit of radius a with a constant angular velocity co . The orbit lies 
in the x-y plane, as shown in Fig. 15.10, with the nucleus at the center. The 
observation direction n is defined by the polar angle and lies in the x-z 
plane. The velocity of the electron is 

\(t) = — €l co a sin (a> t + a) + € 2 co a cos (co t + a) (15.71) 

where a is an arbitrary phase angle. If the electron vanishes at / = 0, the 



[Sect. 15.8] Bremsstrahlung, Virtual Quanta, Radiative Beta Processes 529 




Fig. 15.10 



frequency spectrum of emitted radiation (14.67) is approximately 



dljco) 
dQ. 



e 2 co 2 
4tt 2 c 3 



I ° n x [n x v(0y°* dt 

J — CO 



(15.72) 



where we have assumed that jcoajc) < 1 (dipole approximation) and put 
the retardation factor equal to unity. The integral in (15.72) can be 
written /.„ 

dt = — o> a(€ 1 / 1 + € M cos 0/ 2 ) (15.73) 

J — oo 

where ,.„ 

I x = cos (ay + 0L)e i<ot dt 

J — 00 



-f- 

J —co 



sin jco t + a)e i0,t df 



(15.74) 



and € ± , € are unit polarization vectors perpendicular and parallel to the 
plane containing n and the z axis. The integrals are elementary and lead 
to an intensity distribution, 



dI(co) e 2 co 2 



2^2 



a>n a 



dQ. 4tt 2 c 3 (a) 2 - co 2 f 

X [(o> 2 cos 2 a + w 2 sin 2 a) + cos 2 d(co 2 sin 2 a + co 2 cos 2 a)] (15.75) 

Since the electron can be captured from any position around the orbit, we 
average over the phase angle a. Then the intensity distribution is 

dljco) 
dQ 
The total energy radiated per unit frequency interval is 



i) = _il M* coW + O K1 + cos 2 d) (15 76) 

! 4tt 2 c A c J (co 2 — oo 2 f 



I(fi>) 



— 2 e 2 /co aY 
3rr c \ ci 



' oo\(x> 2 + co 2 )" 
. (a, 2 - co 2 ) 2 J 



(15.77) 



530 



Classical Electrodynamics 



Nftco) 




Fig. 15.11 Spectrum of photons emitted in orbital electron capture because of dis- 
appearance of the charge of the electron. 



while the number of photons per unit energy interval is 

O} 2 ((O 2 + co 2 ) 



*->-£©(?)' 



(co 2 - co 2 ) 



2\2 



1 

hco 



— (15.78) 



For co > co the square-bracketed quantity approaches unity. Then 
the spectrum is a typical bremsstrahlung spectrum. But for co ~ co the 
intensity is very large (infinite in our approximation). The behavior of the 
photon spectrum is shown in Fig. 15.11. The singularity at co = co may 
seem alarming, but it is really quite natural and expected. If the electron 
were to keep orbiting forever, the radiation spectrum would be a sharp line 
at co = co . The sudden termination of the periodic motion produces a 
broadening of the spectrum in the neighborhood of the characteristic 
frequency. 

Quantum mechanically the radiation arises when an / = 1 electron 
(mainly from the 2p orbit) makes a virtual radiative transition to an / = 
state, from which it can be absorbed by the nucleus. Thus the frequency 
co must be identified with the frequency of the characteristic 2p -> \s 
X-ray, hco ~ (3Z 2 e 2 /$a ). Similarly the orbit radius is actually a transitional 
dipole moment. With the estimate a ~ a /Z, where a is the Bohr radius, 
the photon spectrum (15.78) is 

Mto^-Lz^Y-LrNV + «*">' 

2>2tt \hc/ Hco 



L (co 2 - co 2 f 



(15.79) 



[Sect. 15.8] Bremsstrahlung, Virtual Quanta, Radiative Beta Processes 531 

The essential characteristics of this spectrum are its strong peaking at the 
X-ray energy and its dependence on atomic number as Z 2 . 

So far we have considered the radiation which accompanies the disap- 
pearance of the charge of an orbital electron in the electron-capture 
process. An electron possesses a magnetic moment as well as a charge. 
The disappearance of the magnetic moment also gives rise to radiation, 
but with a spectrum of quite different character. The intensity distribution 
in angle and frequency for a point magnetic moment in motion is given by 
(14.74). The electronic magnetic moment can be treated as a constant 
vector in space until its disappearance at t = 0. Then, in the dipole 
approximation, the appropriate intensity distribution is 

dIM = J^L I f ° n x ^ dt 2 (15.80) 

dQ 47rVU-°o 

This gives 

^) = -^L„ 2 sin 2 (15.81) 

dQ 47T 2 c sr 

where is the angle between jjl and the observation direction n. 

In a semiclassical sense the electronic magnetic moment can be thought 
of as having a magnitude fx = V3(ehl2mc), but being observed only 
through its projection p z = ±{eh\2mc) on an arbitrary axis. The moment 
can be thought of as precessing around the axis at an angle a = tan -1 V 2, 
so that on the average only the component of the moment along the axis 
survives. It is easy to show that on averaging over this precession sin 2 
in (15.81) becomes equal to its average value of f , independent of obser- 
vation direction. Thus the angular and frequency spectrum becomes 

dlioy) = _^_/M 2 (1582) 

dQ. 877- 2 cW 2 / 

The total energy radiated per unit frequency interval is 

/(„) = f{^X (15.83) 

2-nc \mc 2 '/ 

while the corresponding number of photons per unit energy interval is 

iV(M = r4--^r 2 (15-84) 

IttHc (mc 2 f 

These spectra are very different in their frequency dependence from a 
bremsstrahlung spectrum. They increase with increasing frequency, 
apparently without limit. Of course, we have been forewarned that our 



532 



Classical Electrodynamics 



classical results are valid only in the low-frequency limit. We can imagine 
that some sort of uncertainty-principle argument such as was used in 
Section 15.7 for radiative beta decay holds here and that conservation of 
energy, at least, is guaranteed. Actually, modifications arise because a 
neutrino is always emitted in the electron-capture process. The probability 
of emission of the neutrino can be shown to depend on the square of its 
energy E v . When no photon is emitted, the neutrino has the full decay 
energy E v = E . But when a photon of energy hco accompanies it, the 
neutrino's energy is reduced to EJ = E — hco. Then the probability of 
neutrino emission is reduced by a factor, 



eA 2 = L _ HcoY 
Ej \ EJ 



(15.85) 



This means that our classical spectra (15.83) and (15.84) must be corrected 
by multiplication with (15.85) to take into account the kinematics of the 
neutrino emission. The modified classical photon spectrum is 



N(hco) = 



ha> 



2\2 



Inhc (mc ) 



. hco 



(15.86) 



This is essentially the correct quantum-mechanical result. A comparison 
of the corrected distribution (15.86) and the classical one (15.84) is shown 
in Fig. 15.12. Evidently the neutrino-emission probability is crucial in 
obtaining the proper behavior of the photon energy spectrum. For the 



N(hu) 




Fig. 15.12 Spectrum of photons emitted in orbital electron capture because of dis- 
appearance of the magnetic moment of the electron. 



[Sect. 15.8] Bremsstrahlung, Virtual Quanta, Radiative Beta Processes 533 



N(fio>) 




hoiQ 



Fig. 15.13 Typical photon spectrum for radiative orbital electron capture with energy 
release E , showing the contributions from the disappearance of the electronic charge 

and magnetic moment. 



customary bremsstrahlung spectra such correction factors are less im- 
portant because the bulk of the radiation is emitted in photons with 
energies much smaller than the maximum allowable value. 

The total radiation emitted in orbital electron capture is the sum of the 
contributions from the disappearance of the electric charge and of the 
magnetic moment. From the different behaviors of (15.79) and (15.86) we 
see that the upper end of the spectrum will be dominated by the magnetic- 
moment contribution, unless the energy release is very small, whereas the 
lower end of the spectrum will be dominated by the electric-charge term, 
especially for high Z. Figure 15.13 shows a typical combined spectrum 
for Z ~ 20-30. Observations on a number of nuclei confirm the general 
features of the spectra and allow determination of the energy release 
E n . 



REFERENCES AND SUGGESTED READING 



Classical bremsstrahlung is discussed briefly by 

Landau and Lifshitz, Classical Theory of Fields, Section 9.4, 
Panofsky and Phillips, Section 19.6. 
A semiclassical discussion, analogous to ours, but much briefer, appears in 
Rossi, Section 2.12. 



534 Classical Electrodynamics 

Bremsstrahlung can be described accurately only by a proper quantum-mechanical 
treatment. The standard reference is 
Heitler. 
The method of virtual quanta (Weizsacker- Williams method) has only one proper 
reference, the classic article by 

Williams. 
Short discussions appear in 
Heitler, Appendix 6, 
Panofsky and Phillips, Section 18.5. 
Among the quantum-mechanical treatments of radiative beta processes, having com- 
parisons with experiment in some cases, are those by 

C. S. W. Chang and D. L. Falkoff, Phys. Rev., 76, 365 (1949), 
P. C. Martin and R. J. Glauber, Phys. Rev., 109, 1307 (1958), 
Siegbahn, Chapter XX (III) by C. S. Wu. 



PROBLEMS 

15.1 A nonrelativistic particle of charge e and mass m collides with a fixed, 
smooth, hard sphere of radius R. Assuming that the collision is elastic, 
show that in the dipole approximation (neglecting retardation effects) the 
classical differential cross section for the emission of photons per unit 
solid angle per unit energy interval is 

d 2 o R* e 2/ y \2 j 

- diU:(2+3sin«e) 



dQ.d(ft<jL>) I2n hc\cj fra> 

where B is measured relative to the incident direction. Sketch the angular 
distribution. Integrate over angles to get the total bremsstrahlung cross 
section. Qualitatively what factor (or factors) govern the upper limit to 
the frequency spectrum ? 

15.2 Two particles with charges q 1 and q 2 and masses m x and m 2 collide under the 
action of electromagnetic (and perhaps other) forces. Consider the angular 
and frequency distributions of the radiation emitted in the collision. 

(a) Show that for nonrelativistic motion the energy radiated per unit 
solid angle per unit frequency interval in the center of mass coordinate 
system is given by 

dI(io, fi) n* 



dn 4t7 2 c 3 



J \rrti m 2 J 

where x = (x 2 — x 2 ) is the relative coordinate, n is a unit vector in the 
direction of observation, and fi = m x m 2 \(m x + m 2 ) is the reduced mass. 

(b) By expanding the retardation factors show that, if the two particles 
have the same charge to mass ratio (e.g., a deuteron and an alpha particle), 



[Probs. 15] Bremsstrahlung, Virtual Quanta, Radiative Beta Processes 
the leading (dipole) term vanishes and the next-order term gives 
dI{co, Q) <*{f£ + qg£t I L-i„ Kn . x)( x x n) dt 2 



535 



dCl 



4^ 



(c) Relate result (6) to the multipole expansion of Sections 9.1-9.3. 
15.3 Two identical point particles of charge q and mass m interact by means of 
a short-range repulsive interaction which is equivalent to a hard sphere of 
radius R in their relative separation. Neglecting the electromagnetic inter- 
action between the two particles, determine the radiation cross section in the 
center of mass system for a collision between these identical particles to the 
lowest nonvanishing approximation. Show that the differential cross section 
for emission of photons per unit solid angle per unit energy interval is 



d 2 a 



d(haj) dQ. 



= §M^ hoi{l 



+ -iWcos 0) - - 2 V 4 (cos 0)] 



where is measured relative to the incident direction. Compare this result 
with that of Problem 15.1 as to frequency dependence, relative magnitude, 
etc. 
15.4 A particle of charge ze, mass m, and nonrelativistic velocity v is deflected in 
a screened Coulomb field, V(r) = Zze 2 e~ ar lr, and consequently emits 
radiation. Discuss the radiation with the approximation that the particle 
moves in a straight-line trajectory past the force center. 

(a) Show that, if the impact parameter is b, the energy radiated per unit 
frequency interval is 



I{o>, b) =- 1 — s 1 1 — 1 a^VO 

3tt c \mc i ] \v/ 



for to < v/b, and negligible for co > vjb. 
(b) Show that the radiation cross section is 



, s 16 ZV/zVV/c 

x(co) ^T—\7n7>)\v 



2r 



■(k \x)-KJ 



\x) + 



IK^Kjix) 



Jx 1 



where X x = a&min, X 2 = 0C& m ax. 

(c) With bmm = fi/mv, 6 max = i?/o>, and or 1 = l.4a Z~^, determine the 
radiation cross section in the two limits, x 2 <^l an(i *2 ^ * • Compare your 
results with the "screening" and "no screening" limits of the text. 
15.5 A particle of charge ze, mass m, and velocity v is deflected in a hyperbolic 
path by a fixed repulsive Coulomb potential, V(r) = Zze 2 jr. In the non- 
relativistic dipole approximation (but with no further approximations), 

(a) show that the energy radiated per unit frequency interval by the 
particle when initially incident at impact parameter b is : 



8 (zeaco) 2 , , 



' o) (h^Q_ 



3tt c 3 

(b) show that the radiation cross section is 
16 {zeavf 



+ 



- 1 



K. 



i(o/ca 



Z(-)= T -^ 



g— (JTOJ/CO 



CO / 0)\ , I U>\ 

a>o Wo/ L \ w o/ 



536 Classical Electrodynamics 

(c) Prove that the radiation cross section reduces to that obtained in the 
text for classical bremsstrahlung for co < a> . What is the limiting form 
fora>>co ? b 

(d) What modifications occur for an attractive Coulomb interaction? 
The hyperbolic path may be described by 

x = a( e + cosh £), y = -b sinh £, to t = (f + e sinh f) 
where a = Zze 2 /mv 2 , e = Vl + (b/a) 2 , co = vja. 
15.6 Using the method of virtual quanta, discuss the relationship between the 
cross section for photodisintegration of a nucleus and electrondisintegration 
of a nucleus. 

(a) Show that, for electrons of energy E = ymc 2 > mc 2 , the electron- 
disintegration cross section is approximately: 

where hw> T is the threshold energy for the process. 

(b) Assuming that ff photo (eo) has the resonance shape: 

A e 2 r 

"photoC") ^ Yn Mc (co-co^+cr^ 

where the width T is small compared to (co - co T ), sketch the behavior of 
o e i(E) as a function of E and show that for E > fico , 



a el (E) ~ - (-) — 1 In / ^ 2 \ 

v \HcJ Mc co \mc 2 ho) J 



(c) Discuss the experimental comparison between activities produced by 
bremsstrahlung spectra and monoenergetic electrons as presented by 
Brown and Wilson, Phys. Rev., 93, 443 (1954), and show that the quantity 
defined as F e x P (Z, E) has the approximate value 8tt/3 at high energies if the 
Weizsacker- Williams spectrum is used to describe both processes and the 
photodisintegration cross section has a resonance shape. 
15.7 A fast particle of charge ze, mass M, velocity v, collides with a hydrogen-like 
atom with one electron of charge — e, mass m, bound to a nuclear center of 
charge Ze. The collisions can be divided into two kinds: close collisions 
where the particle passes through the atom (b < d), and distant collisions 
where the particle passes by outside the atom (b > d). The atomic "radius" 
d can be taken as a /Z. For the close collisions the interaction of the 
incident particle and the electron can be treated as a two-body collision and 
the energy transfer calculated from the Rutherford cross section. For the 
distant collisions the excitation and ionization of the atom can be considered 
the result of the photoelectric effect by the virtual quanta of the incident 
particle's fields. 

For simplicity assume that for photon energies Q greater than the 
ionization potential / the photoelectric cross section is 



[Probs. 15] Bremsstrahlung, Virtual Quanta, Radiative Beta Processes 537 

(This obeys the empirical Z 4 A 3 law for X-ray absorption and has a coefficient 
adjusted to satisfy the dipole sum rule, J o v (Q) dQ = l^e^/mc.) 

(a) Calculate the cross sections for energy transfer Q for close and 
distant collisions (write them as functions of Q\l as far as possible and in 
units of iTTZ^Imv 2 ^). Plot the two distributions for Qjl > 1 for non- 
relativistic motion of the incident particle and \mv 2 = 10 3 /. 

(b) Show that the number of distant collisions is much larger than the 
number of close collisions, but that the energy transfer per collision is 
much smaller. Show that the energy loss is divided approximately equally 
between the two kinds of collisions, and verify that your total energy loss 
is in essential agreement with Bethe's result (13.44). 

15.8 In the decay of a pi meson at rest a mu meson and a neutrino are created. 
The total kinetic energy available is (m„ — m^c 2 = 34 Mev. The mu 
meson has a kinetic energy of 4.1 Mev. Determine the number of quanta^ 
emitted per unit energy interval because of the sudden creation of the 
moving mu meson. Assuming that the photons are emitted perpendicular 
to the direction of motion of the mu meson (actually it is a sin 2 distribu- 
tion), show that the maximum photon energy is 17 Mev. Find how many 
quanta are emitted with energies greater than one-tenth of the maximum, 
and compare your result with the observed ratio of radiative pi-mu decays. 
[W. F. Fry, Phys. Rev., 86, 418 (1952); H. Primakoff, Phys. Rev., 84, 1255 
(1951).] 

15.9 In internal conversion, the nucleus makes a transition from one state to 
another and an orbital electron is ejected. The electron has a kinetic energy 
equal to the transition energy minus its binding energy. For a conversion 
line of 1 Mev determine the number of quanta emitted per unit energy 
because of the sudden ejection of the electron. What fraction of the electrons 
will have energies less than 99 per cent of the total energy? Will this 
low-energy tail on the conversion line be experimentally observable? 



16 



Multipole Fields 



In Chapters 3 and 4 on electrostatics the spherical harmonic 
expansion of the scalar potential was used extensively for problems 
possessing some symmetry property with respect to an origin of coordinates. 
Not only was it useful in handling boundary-value problems in spherical 
coordinates, but with a source present it provided a systematic way of 
expanding the potential in terms of multipole moments of the charge 
density. For time-varying electromagnetic fields the scalar spherical 
harmonic expansion can be generalized to an expansion in vector spherical 
waves. These vector spherical waves are convenient for electromagnetic 
boundary-value problems possessing spherical symmetry properties and for 
the discussion of multipole radiation from a localized source distribution. 
In Chapter 9 we have already considered the simplest radiating multipole 
systems. In the present chapter we present a systematic development. 



16.1 Basic Spherical Wave Solutions of the Scalar Wave Equation 

As a prelude to the vector spherical wave problem, we consider the 
scalar wave equation. A scalar field ^(x, t) satisfying the source-free 
wave equation, 2 

V V -i|| = (16.1, 

can be Fourier-analyzed in time as 

/*oo 

ip(x, = y(x, (o)e- i(at da) (16.2) 

J — oo 

with each Fourier component satisfying the Helmholtz wave equation, 

(V 2 + k 2 )y)(x, co) = (16.3) 

538 



[Sect. 16.1] 



Multipole Fields 



539 



with k 2 = co 2 jc 2 . For problems possessing symmetry properties about 
some origin it is convenient to have fundamental solutions appropriate to 
spherical coordinates. The representation of the Laplacian operator in 
spherical coordinates is given in equation (3.1). The separation of the 
angular and radial variables follows the well-known expansion, 



<Kx, co) = 2 Mr)Y lm (0, <f>) 



(16.4) 



where the spherical harmonics Y lm are defined by (3.53). The radial 
functions / f (r) satisfy the radial equation, 



• d 2 2 d 2 /(/ + 1) ' 
Jr 2 rdr r 2 . 



Mr) = 



With the substitution, 



fi(r) = -rz u t (r) 



equation (16.5) is transformed into 



d 2 Id ; , 2 (/ + |) 2 ' 
Ur 2 rdr r 2 . 



»i(r) = 



(16.5) 



(16.6) 



(16.7) 



This equation is just Bessel's equation (3.75) with v = I + \. Thus the 
solutions for fir) are 

(16.8) 



fir) ~ -£■ J t+ i4(kr), -% N l+l4 (kr) 



It is customary to define spherical Bessel and Hankel functions, denoted 
by ;*(*)> «i(«), ti?' 2) {x), as follows: 



nix) = \^A N l+ i A (x) 
h?- 2 \x) = (£) [_J l+ y 2 (x) ± iN l + H (x)] 



(16.9) 



For real x, h\ 2) (x) is the complex conjugate of h^ix). From the series 
expansions (3.82) and (3.83) one can show that 



\a; dxl \ x I 

n x {x)= -(-*)M- — I ) 

\x dxl \ x ! 



(16.10) 



540 Classical Electrodynamics 

For the first few values of / the explicit forms are : 



JoO) 



sin a; 



n (x) = 



cos a; 



IX 



i ( T \ - !iBi? cosx „ r„\ cos a; sin x 

h?\x) = - e -(i + '-) 

x \ xl 
j&) = (\ ~ -) sin x _ Icosf ^ {x) = _ /3 l\ cos x _ sin* 

^ xl X 2 \x 3 Xl x 2 



a;- \ x x £ I 



(16.11) 

From the asymptotic forms (3.89)-(3.91) it is evident that the small 
argument limits are 



x </ 



Ji( x ) 



(21 + 1)! ! 



"*(*) -> - 



(21 - 1)! ! 



(16.12) 



where (2/ + 1)! ! = (21 + 1)(2/ - 1)(2/ - 3) • • • (5) • (3) • (1). Similarly 
the large argument limits are 



x >/ 



Ji(x)->- sin [x-— I 

x \ 2/ 

n t (x) -+ cos he 

x \ 2/ 



(16.13) 



The spherical Bessel functions satisfy the recursion formulas 

2/ + 1 

— — z t (x) = z^x) + z l+1 (x) 

1 



Zl ' (x) = JTTi V z i-^ - V + i>i+i(*>D 



(16.14) 



[Sect. 16.1] Multipole Fields 541 

where z t (x) is any one of the functions j\(x), «,(*), htpix), h\ 2) (x). The 
Wronskians of the various pairs are 

W(h n,) = - W(j t1 h™) = - W(n lt h™) = \ (16.15) 

The general solution of (16.3) in spherical coordinates can be written 
Vtt = 2 lAWXkr) + A^hf\kr)-\Y lm {d, <f>) (16.16) 

where the coefficients A\H and 4}^ will be determined by the boundary 
conditions. 

For reference purposes we present the spherical wave expansion for the 
outgoing wave Green's function G(x, x'), which is appropriate to the 
equation, 

(V 2 + k 2 )G(x, x') = -<3(x - x') (16.17) 

in the infinite domain. The closed form for this Green's function, as was 
shown in Chapter 9, is 

ik\x-*\ 

G(x,x') = -^ (16.18) 

47t\x — x I 

The spherical wave expansion for G(x, x') can be obtained in exactly the 
same way as was done in Sections 3.8 and 3.10 for Poisson's equation 
[see especially equation (3.117) and below, and (3.138) and below]. An 
expansion of the form, 

G(x, x') = 2 gfr, r')Y* m (0', <f>')Y lm (d, <f>) (16.19) 

l,m 

substituted into (16.17) leads to an equation for g t (r, r'): 



il + I ±L + k 2 _ KijtAi 



dr 2 r dr r 2 



U = ~ - 2 Kr ~ r') (16.20) 
r 2 



The solution which satisfies the boundary conditions of finiteness at the 
origin and outgoing waves at infinity is 

gl (r, r') = AUk r< )h^(kr>) (16.21) 

The correct discontinuity in slope is assured if A = ik. Thus the expansion 
of the Green's function is 

i]c\x—x'\ °°. I 

-1 = ik y j l (/cr<)^ 1) (/cr>) > Y*Jd', <f>')Y lm (6, <f>) (16.22) 

4tt x — x r-1 t- 1 , 



542 



Classical Electrodynamics 



Our emphasis so far has been on the radial functions appropriate to 
the scalar wave equation. We now re-examine the angular functions in 
order to introduce some concepts of use in considering the vector wave 
equation. The basic angular functions are the spherical harmonics 
Yim(Q> 4) (3-53), which are solutions of the equation, 



1 



1 



.sin dd \ 30/ sin 2 d<f> 2 



Y lm =l{l + \)Y lm (16.23) 



As is well known in quantum mechanics, this equation can be written in 
the form : 

L 2 Y lm = l{l+\)Y ln (16.24) 

The differential operator L 2 = L 2 + L v 2 + L 2 , where 



L = - (r x V) 



(16.25) 



is the orbital angular-momentum operator of wave mechanics. 
The components of L can be written conveniently in the combinations, 



L + =L X + iL y = <?*(— + i cot — 
+ x \do dcf> 

L_ = L x - iL v = e-'+l- — + / cot — 
\ dd dcf> 

r • d 

defy 



(16.26) 



We note that L operates only on angular variables and is independent of r. 
From definition (16.25) it is evident that 



r-L = 



(16.27) 



holds as an operator equation. From the explicit forms (16.26) it is easy 
to verify that L 2 is equal to the operator on the left side of (16.23). 

From the explicit forms (16.26) and recursion relations for Y lm the 
following useful relations can be established: 



L + Y lm = V(l - m)(l + m + \)Y h 



L_Y lm = V(l + m)(l-m + l)Y l: 



w— 1 



L K Y, m = mYr 



(16.28) 



[Sect. 16.2] Multipole Fields 543 

Finally we note the following operator equations concerning the com- 
mutation properties of L, L 2 , and V 2 : 

L 2 L = LL 2 1 

LxL = /L ► (16.29) 

L,.V 2 = V 2 L, 
where 

(16.30) 



V 2 = -—Ar)-- 



rdr 2 



16.2 Multipole Expansion of the Electromagnetic Fields 

In a source-free region Maxwell's equations are 

c dt c ot 



(16.31) 



V • E = 0, V • B = 

With the assumption of a time dependence, e~ mt , these equations become 
VxE = iB, V x B = - ikE 
V • E = 0, V.B = 



(16.32) 



If E is eliminated between the two curl equations, we obtain the following 
equations, 

(V 2 + fc 2 )B = 0, V • B = " 
and the defining relation, 



E = -V xB 

k 

Alternatively B can be eliminated to yield 

(V 2 + fc 2 )E = 0, V • E = 



(16.33) 



plus 



B = — V xE 



(16.34) 



Either (16.33) or (16.34) is a set of three equations which is equivalent to 
Maxwell's equations (16.32). 

We now wish to determine multipole solutions for E and B. From 
(16.33) it is evident that each rectangular component of B satisfies the 
Helmholtz wave equation (16.3). Hence each component of B can be 



544 Classical Electrodynamics 

represented by the general solution (16.16). These can be combined to 
yield the vectorial result : 

B = 1 [AgftPtyr) + A£h«>(fcr)]y IfII (0, <j>) (16.35) 

l ,m 

where A lm are arbitrary constant vectors. 

The coefficients A lm in (16.35) are not completely arbitrary. The 
divergence condition V • B = must be satisfied. Since the radial 
functions are linearly independent, the condition V • B = must hold for 
the two sets of terms in (16.35) separately. Thus we require the coefficients 
A lm to be so chosen that 

V • ^ hflcrfr^JO, <f>) = (16.36) 

l,m 

The gradient operator can be written in the form : 



r dr 



V = -- -r xL (16.37) 



where L is the operator (16.25). When this is applied in (16.36), we obtain 
the requirement, 

r • 2 [f: 2 Alm Ylm - 7 1 L x 2 AimYlm ] = ° (1638) 

I m m 

From recursion formulas (16. 14) it is evident that in general the coefficients 
A lm for a given / will be coupled with those for /' = / ± 1. This will 
happen unless the (2/ + 1) vector coefficients for each / value are such that 

r-lA ( J Im = (16.39) 

TO 

For this special circumstance, the second term in (16.38) shows that the 
final condition on the coefficients is 

r-(L xZA lm Y l J = (16.40) 

TO 

The assumption (16.39) that the field is transverse to the radius vector, 
together with (16.40), is sufficient to determine a unique set of vector 
angular functions of order /, one for each m value. These can be found in 
a straightforward manner from (16.39) and (16.40), and the properties of 
the Y lm 's. But it is expedient, and not too damaging at this point, to 
observe that the appropriate angular solution is 

2 A lm' Y lm' = 2 ^ ln LY lm (16.41) 

to' m 

From (16.27) it is clear that the transversality condition (16.39) is satisfied. 
Similarly, from the second commutation relation in (16.29) and (16.27), 



[Sect. 16.2] Multipole Fields 545 

the final condition (16.40) is obeyed. That the functions [/i(r)IT im ] 
satisfy the wave equation (16.3) follows from the last commutation 
relation in (16.29). 

By assumption (16.39) we have found a special set of electromagnetic 
multipole fields, 

B Im =/,(fcr)Ly Im (M)) 



Ej m — T v X B lm 
k 



(16.42) 

Kim = ~ V X °lm J 

where 

f l{ kr) = APh?\kr) + AfWfXkr) (16.43) 

Any linear combination of these fields, summed over / and m, satisfies the 
set of equations (16.33). They have the characteristic that the magnetic 
induction is perpendicular to the radius vector (r • B lm = 0). They 
therefore do not represent a general solution to equations (16.33). They 
are, in fact, the spherical equivalent of the transverse magnetic (TM), or 
electric, cylindrical fields of Chapter 8. 

If we had started with the set of equations (16.34) instead of (16.33), we 
would have obtained an alternative set of multipole fields in which E is 
transverse to the radius vector: 

K lm =Mkr)LY lm (0,cf>)) 

-»« T, ( 1644 ) 

B Jro = — V x E fTO J 

These are the spherical wave analogs of the transverse electric (TE), or 
magnetic, cylindrical fields of Chapter 8. 

Just as for the cylindrical wave-guide case, the two sets of multipole 
fields (16.42) and (16.44) can be shown to form a complete set of vector 
solutions to Maxwell's equations. The terminology electric and magnetic 
multipole fields will be used, rather than TM and TE, since the sources of 
each type of field will be seen to be the electric-charge density and the 
magnetic-moment density, respectively. Since the vector spherical 
harmonic, L Y lm , plays an important role, it is convenient to introduce the 
normalized form,* 

x lm (d, <f>) = ,-i — Lr Jm (0, <f>) (16.45) 

V/(/ + i) 

with the orthogonality property, 

X? m ..X lm <*Q = Mmm- (16-46) 



/■ 



* Xi m is defined to be identically zero for / = 0. Spherically symmetric solutions to 
the source-free Maxwell's equations exist only in the static limit k -* 0. 



546 Classical Electrodynamics 

By combining the two types of fields we can write the general solution to 
Maxwell's equations (16.32): 



B = ^ [a E (l, m)fflcr)X lm - l - a M (l, m)V x g,(fcr)X,. 

l,m 

E = ^ Mr a E (l, m)V x /j(/cr)X ?m + a M (l, m^fciOX, 



(16.47) 



where the coefficients a E {l, m) and a M (I, m) specify the amounts of electric 
(/, m) multipole and magnetic (/, m) multipole fields. The radial functions 
fi(kr) andg;(&r) are of form (16.43). The coefficients a E (l, m) and a M (l, m), 
as well as the relative proportions in (16.43), will be determined by the 
sources and boundary conditions. 



16.3 Properties of Multipole Fields; Energy and Angular Momentum 
of Multipole Radiation 

Before considering the connection between the general solution (16.47) 
and a localized source distribution, we examine the properties of the 
individual multipole fields (16.42) and (16.44). In the near zone (kr < 1) 
the radial function f t (kr) is proportional to n^ given by (16.12), unless its 
coefficient vanishes identically. Excluding this possibility, the limiting 
behavior of the magnetic induction for an electric (/, m) multipole is 

B lw --yL^ (16.48) 

where the proportionality coefficient is chosen for later convenience. To 
find the electric field we must take the curl of the right-hand side. A 
useful operator identity is 

7 2 «/. . ^ 



iV x L = rV 2 - VI 1 + r — I (16.49) 

The electric field (16.42) is 

E,.-yVxL(&) (16.50) 

Since (Y lm lr l+1 ) is a solution of Laplace's equation, the first term in 
(16.49) vanishes. The second term merely gives a factor /. Consequently 
the electric field at close distances for an electric (/, m) multipole is 






E lm -*-V ^ (16.51) 



[Sect. 16.3] Multipole Fields 547 

This is exactly the electrostatic multipole field of Section 4.1. We note 
that the magnetic induction B lm is smaller in magnitude than E lm by a 
factor kr. Hence, in the near zone, the magnetic induction of an electric 
multipole is always much smaller than the electric field. For the magnetic 
multipole fields (16.44) evidently the roles of E and B are interchanged 
according to the transformation, 

E S ^-B M , B B ^E M (16.52) 

In the far or radiation zone {kr > 1) the multipole fields depend on the 
boundary conditions imposed. For definiteness we consider the example 
of outgoing waves, appropriate to radiation by a localized source. Then 
the radial function f t (kr) is proportional to the spherical Hankel function 
hl x) {kr). From the asymptotic form (16.13) we see that in the radiation 
zone the magnetic induction for an electric (/, m) multipole goes as 

B^->(-*y +1 f- r L7 lrn (16.53) 

kr 



Then the electric field can be written 

(-0* 



E7™ = 



ikr\ p ikr 

V\ e —J xLY lm + ^-V xLY t1 



(16.54) 



Since we have already used the asymptotic form of the spherical Hankel 
function, we are not justified in keeping higher powers in (1/r) than the 
first. With this restriction and use of the identity (16.49) we find 



E Jm = -(-0 l+1 f-" n x LY JW - i (rV 2 - V)y If 
kr L k 



(16.55) 



where n = (r/r) is a unit vector in the radial direction. The second term is 
evidently \\kr times some dimensionless function of angles and can